Algebra Function Basic Flashcards
In algebra, how is a function defined in simplest terms?
A rule that assigns each input exactly one output.
What is the ‘domain’ of a function f?
All possible input values (x-values) for which f(x) is defined.
What is the ‘codomain’ of a function?
The set in which all outputs of the function are allowed to lie.
What is the ‘range’ (or image) of a function?
All actual output values the function produces from its domain.
What does it mean for a relation to ‘fail the vertical line test’?
At least one vertical line intersects the graph in more than one point, so it is not a function.
Define a ‘one-to-one’ (injective) function.
A function where each output is produced by at most one input (distinct inputs → distinct outputs).
Define an ‘onto’ (surjective) function.
A function whose range equals its entire codomain (every element in the codomain is mapped from some input).
What does ‘bijective’ mean for a function?
It is both injective and surjective, so it has a perfect one-to-one correspondence between domain and codomain.
In function notation, f: A → B, what do A and B represent?
A is the domain, B is the codomain.
What is the difference between ‘range’ and ‘codomain’?
Range is the actual set of outputs. Codomain is the set from which outputs can potentially come.
How do you typically find the domain of a function given by an expression?
Identify values of x that make the expression undefined or invalid (e.g., dividing by zero, negative radicands for even roots), then exclude them.
How do you find the domain of a rational function f(x) = P(x)/Q(x)?
Exclude values for which Q(x) = 0 from the real domain.
What is the domain of f(x) = √(x - 3) in real numbers?
All x ≥ 3.
What is the domain of f(x) = 1/(x + 2) in real numbers?
All real x except x ≠ -2.
If f(x) = √(x + 4) + (1/(x - 2)), how do you determine domain?
First x + 4 ≥ 0 → x ≥ -4, and x - 2 ≠ 0 → x ≠ 2. Combine to get x ≥ -4 but x ≠ 2.
Give an example of a function with domain all real numbers ℝ.
Any polynomial, e.g., f(x) = x² + 1.
How do you typically find the range of a function algebraically?
Solve y = f(x) for x in terms of y, then determine permissible y-values (sometimes more advanced).
What is the range of the function f(x) = x², x ∈ ℝ?
All real y ≥ 0.
What is the range of f(x) = 2x + 3, x ∈ ℝ?
All real numbers (−∞, ∞).
For f(x) = 1/x, x ≠ 0, what is its range in real numbers?
All real y except y ≠ 0.
Define a linear function in one variable.
A function of the form f(x) = mx + b, where m and b are constants.
Define a constant function.
A function where f(x) = c for all x in the domain, c is a constant.
What is a polynomial function?
A function f(x) = aₙxⁿ + … + a₁x + a₀ where coefficients aᵢ are real numbers and exponents are nonnegative integers.
Define a quadratic function.
A polynomial function of degree 2: f(x) = ax² + bx + c, with a ≠ 0.
What is a cubic function?
A polynomial function of degree 3, e.g. f(x) = ax³ + bx² + cx + d.
What is a rational function?
A function that can be written as the ratio of two polynomials, f(x) = P(x)/Q(x).
Define a piecewise function.
A function defined by different expressions on different intervals of its domain.
Define an exponential function (with base b>0, b≠1).
f(x) = b^x, domain = ℝ, range = (0, ∞).
Define a logarithmic function (base b>0, b≠1).
f(x) = log_b(x), domain = (0, ∞), range = ℝ.
What is an absolute value function in standard form?
f(x) = |x|, which outputs the nonnegative magnitude of x.
What is the signum (sign) function?
sgn(x) = 1 if x>0, 0 if x=0, and -1 if x<0.
What is the greatest integer (floor) function?
⌊x⌋ gives the largest integer less than or equal to x.
What is the fractional part function?
{x} = x - ⌊x⌋, the ‘decimal part’ of x.
Define the step function f(x)=c for intervals, e.g., the Heaviside step function.
Heaviside: H(x)=0 if x<0, 1 if x≥0. It’s a piecewise constant function.
What is the identity function on ℝ?
f(x) = x for all x in ℝ.
What is the zero function on ℝ?
f(x) = 0 for all x in ℝ.
Define an ‘even’ function in terms of symmetry.
f(−x) = f(x) for all x in domain; symmetric about the y-axis.
Define an ‘odd’ function in terms of symmetry.
f(−x) = −f(x) for all x in domain; symmetric about the origin.
Give an example of an even function.
f(x) = x².
Give an example of an odd function.
f(x) = x³.
What is the vertical line test for a function’s graph?
Any vertical line should intersect the graph at most once for it to be a function.
Define a horizontal shift for a function f(x).
g(x) = f(x - h) shifts f(x) h units to the right if h>0, left if h<0.
Define a vertical shift for a function f(x).
g(x) = f(x) + k shifts the graph k units up if k>0, down if k<0.
Define a horizontal scaling for a function f(x).
g(x) = f(bx). If |b|>1, it compresses horizontally; if 0<|b|<1, it stretches.
Define a vertical scaling for a function f(x).
g(x) = a·f(x). If |a|>1, vertical stretch; if 0<|a|<1, vertical shrink.
How do you reflect a function across the x-axis?
Use g(x) = −f(x).
How do you reflect a function across the y-axis?
Use g(x) = f(−x).
If y=f(x) is known, what function form flips the graph over the line y=x (i.e., reflection across y=x)?
The inverse function, x = f(y) solved for y, or y = f⁻¹(x).
How is the graph of y=f(x)+3 derived from y=f(x)?
Shift it up by 3 units.
How is the graph of y=f(x−2) derived from y=f(x)?
Shift it right by 2 units.
Define the sum of two functions f and g.
(f + g)(x) = f(x) + g(x).
Define the difference of two functions f and g.
(f − g)(x) = f(x) − g(x).
Define the product of two functions f and g.
(fg)(x) = f(x)·g(x).
Define the quotient of two functions f and g, g≠0.
(f/g)(x) = f(x)/g(x), domain excludes where g(x)=0.
Define the composition (f ∘ g)(x).
f(g(x)). First apply g, then apply f to the result.
What is an example of function composition usage?
If f(x)=2x+1 and g(x)=x², then (f ∘ g)(x)=f(x²)=2x²+1.
How do you find the domain of (f ∘ g)(x)?
Include only x values in the domain of g for which g(x) is in the domain of f.
What property must hold for function composition to make sense?
The range of g must overlap with the domain of f.
Is function composition generally commutative?
No; (f ∘ g)(x) usually differs from (g ∘ f)(x).
If h(x) = (f + g)(x), how is this typically simplified?
h(x)=f(x)+g(x). Keep domain constraints in mind from both f and g.
Define an inverse function f⁻¹ if f is one-to-one.
A function that undoes f, satisfying f⁻¹(f(x))=x and f(f⁻¹(x))=x.
What is the key property that allows a function to have an inverse?
The function must be injective (one-to-one).
What is the graphical relationship of a function and its inverse?
They are reflections of each other across the line y=x.
How do you find an inverse function algebraically?
1) Write y=f(x). 2) Solve for x in terms of y. 3) Switch x and y, get y=f⁻¹(x).
What is the domain of f⁻¹ compared to f?
It is the range of f.
What is the range of f⁻¹ compared to f?
It is the domain of f.
If f(x)=2x+5, find f⁻¹(x).
f⁻¹(x)=(x−5)/2.
For f(x)=x³, find f⁻¹(x).
f⁻¹(x)=³√x.
If f(x)=|x|, is it invertible over ℝ?
No (it fails injectivity). Usually restricted to x≥0 or x≤0 for an inverse.
Is the function f(x)=x² invertible on all real ℝ?
No (it’s not one-to-one). Restrict domain to x≥0 or x≤0 to define an inverse.
Define ‘increasing function’ on an interval.
f(x₁) < f(x₂) for all x₁ < x₂ in that interval.
Define ‘decreasing function’ on an interval.
f(x₁) > f(x₂) for all x₁ < x₂ in that interval.
Define a constant function on an interval.
f(x₁) = f(x₂) for all x₁, x₂ in that interval.
What is a local maximum of a function?
A point where f(x₀) ≥ f(x) for x near x₀.
What is a local minimum of a function?
A point where f(x₀) ≤ f(x) for x near x₀.
Define a continuous function at x=a.
f is continuous at a if lim(x→a) f(x)=f(a).
Define a discontinuous function at x=a.
A function that breaks the continuity condition (limit doesn’t exist, or doesn’t equal f(a)).
Explain boundedness of a function on an interval.
It’s bounded if there’s a real M with |f(x)| ≤ M for all x in that interval.
Define the zero (or root) of a function.
An x-value such that f(x)=0.
What is the difference between local and global extrema of a function?
Local is highest/lowest in a neighborhood; global is highest/lowest over the entire domain.
What is a piecewise-defined function? Give a simple example.
Defined by multiple rules on different intervals. E.g. f(x)=x if x≥0, −x if x<0.
When combining piecewise functions, how do you handle domain overlap?
Match intervals carefully and define sums/products piecewise as well.
Explain function iteration, e.g., f²(x).
f²(x) means f(f(x)), applying f to x twice.
What is the functional equation approach in basic form?
It’s an equation like f(x+y)=f(x)+f(y), used to deduce function forms (like linear for additive).
Define the identity function i(x) on domain D.
i(x)=x for all x ∈ D.
Describe how to reflect a function across the line y=k.
Subtract from k: new function is 2k−f(x).
Describe how to reflect a function across the line x=h.
Replace x with 2h−x in the function rule.
If f is one-to-one, how many times will a horizontal line intersect its graph?
At most once (the horizontal line test).
If f is onto a set B, what does that mean about the range of f?
Range = B exactly.
Why is f(x)=x² not injective on ℝ?
f(−1)=f(1)=1, so multiple x’s produce the same output.
Replace x with 2h−x in the function rule.
For a cost function C(q)=fixed cost + variable cost·q, what is the domain?
q≥0 (assuming no negative quantity).
If f(t)=distance traveled at time t, what is typically the domain?
t≥0, as negative time is not used (depending on context).
In a supply/demand function, p=f(q), domain is q≥0. Why?
Quantity q can’t be negative.
If f(x)=celcius from Fahrenheit, domain is all reals. The formula is f(F)=(5/9)(F−32). Range?
All real Celcius values: (−∞, ∞).
A parent function might be g(x)=x². Then h(x)=x²+3 is a vertical shift. Which direction by how much?
Up 3 units.
An inverse variation model is f(x)=k/x. If x doubles, how does f change?
f is halved, since f(x)∝1/x.
If f is even, then the graph is symmetrical about which axis?
The y-axis.
If f is odd, then f(0)=0 is typical. Why?
Because f(−0)=−f(0) implies f(0)=−f(0) → f(0)=0.
If a function passes the horizontal line test, what property does it have?
It is one-to-one (injective).
Give a direct variation function formula that passes through (0,0) with slope 5.
f(x)=5x.
What is a function transformation in algebra?
An operation (shifting, scaling, reflecting) that modifies a function’s graph into a new position or shape.
Write the general form for a transformed function from a parent f(x).
y = a·f(b(x−h)) + k (combining shift, scale, reflection).
In y = f(x) + k, how does k affect the graph?
It shifts the graph vertically: up by k if k>0, down if k<0.
In y = f(x−h), how does h affect the graph?
It shifts the graph horizontally: right by h if h>0, left if h<0.
For a horizontal shift, is it always x−h in the argument if we move right by h?
Yes. Right shift: x−h with h>0. Left shift: x−(−h)=x+h.
What does y = −f(x) do to the parent function’s graph?
Reflects it across the x-axis.
What does y = f(−x) do to the parent function’s graph?
Reflects it across the y-axis.
Define a vertical stretch by factor a>1.
y = a·f(x), multiplies all y-values by a (makes the graph ‘taller’).
Define a vertical compression by factor 0 < a < 1.
y = a·f(x), reduces all y-values, making the graph ‘flatter’.
How is a horizontal compression by factor b>1 expressed?
y = f(bx), compresses along x-axis by 1/b.
Write the formula for shifting a function f(x) up 4 units.
y = f(x) + 4.
Write the formula for shifting a function g(x) down 2 units.
y = g(x) − 2.
Write the formula for shifting h(x) right 7 units.
y = h(x−7).
Write the formula for shifting p(x) left 5 units.
y = p(x + 5).
Explain how the domain changes when shifting horizontally by h.
Domain’s x-values shift by −h, but the shape remains the same. If original domain is D, new domain is D shifted by h.
If you have y=f(x)+k, does the domain change from the original function f(x)?
No. Vertical shifts do not affect x-values or domain.
In y = (x−3)² + 4, what transformations from y=x² are present?
Right shift by 3, and up shift by 4.
If a function’s domain was x≥0, then we apply x→x−2 shift, what is the new domain?
x−2≥0 → x≥2.
If y=f(x) is known, how do we shift it left by 2 and down by 3 simultaneously?
y = f(x+2) − 3.
Give a single transformation formula for shifting a function f(x) up k units and right h units.
y = f(x−h) + k.
What is the formula for reflection across the x-axis of y=f(x)?
y = −f(x).
What is the formula for reflection across the y-axis of y=f(x)?
y = f(−x).
Which reflection changes sign of y-values: x-axis or y-axis reflection?
x-axis reflection changes y-values’ signs.
Which reflection changes sign of x-values in the function argument?
y-axis reflection uses f(−x).
How do you reflect across y=k (a horizontal line)?
First shift the function so line y=k becomes x-axis, reflect, then shift back: y=k−(f(x)−k).
How do you reflect across x=h (a vertical line)?
Replace x with 2h−x. So y = f(2h−x).
Is reflection across y=x the same as inverse function?
Yes, graphically it’s reflection over y=x, but it only forms an inverse if f is one-to-one.
How do you reflect f(x)=√x across y-axis?
Use y=f(−x)=√(−x). But domain changes to x≤0.
Give the transformation form for reflecting across x-axis and shifting up k.
y = −f(x) + k.
If y=f(x) has domain x≥−1, after reflection across x-axis, does domain change?
No, reflection across x-axis doesn’t affect x, so domain remains x≥−1.
Write a function that vertically stretches f(x) by factor 3.
g(x) = 3f(x).
Write a function that vertically compresses f(x) by factor ½.
g(x) = ½f(x).
If y=f(x) is multiplied by a factor a<0, is that also a reflection?
Yes. a negative factor includes reflection across x-axis plus vertical stretch/compression.
Compare the transformation y=2f(x) vs. y=f(2x) in plain English.
y=2f(x) is vertical stretch by factor 2; y=f(2x) is horizontal compression by factor 1/2.
Which formula changes outputs to half their size: y=0.5f(x) or y=f(0.5x)?
y=0.5f(x) compresses vertically by ½.
If f(x)≥0 for x≥0, then 3f(x)≥?
≥0 as well, scaled up 3 times but no sign change.
In a vertical stretch by factor a>1, does domain or range expand?
Range is stretched, domain is unaffected.
What if y=−2f(x)? Summarize the transformations.
Reflection across x-axis, then vertical stretch by factor 2.
If parent function is y=|x|, what is y=3|x| doing?
Vertical stretch by factor 3.
If y=f(x) passes (2,4), where does y=4f(x) pass for x=2?
At (2,16).
Write a function for horizontally stretching f(x) by factor 2.
g(x)=f(x/2) (the inside is x/2 for a 2× horizontal stretch).
Write a function for horizontally compressing f(x) by factor ½.
g(x)=f(x/(1/2))=f(2x).
Which transformation is y=f(3x)?
A horizontal compression by factor 1/3.
If f(x)=x², then g(x)=f(2x) means g(x)=(2x)²=4x². Which transformation is that?
Horizontal compression by factor ½, but it looks like a vertical stretch by 4 in the equation.
If h>1, what does y=f(hx) do?
Horizontal compression by factor 1/h.
If 0<h<1, what does y=f(hx) do?
Horizontal stretch by factor 1/h (which is >1).
Does y=f(bx) for b>0 change the domain set for real x?
No, but the function’s features appear ‘compressed or stretched’ horizontally.
If y=f(½x), how is the graph changed from y=f(x) at x=4?
We evaluate f(2), so the features shift to the right (it’s a stretch factor 2).
General formula for a horizontal scale factor k>0 is y=f((1/k)x). True or false?
True. If k>1, that’s a stretch; if 0<k<1, a compression.
In a horizontal stretch by factor 3, a point at x=2 moves to x=6 or x=2/3?
It moves to x=6 (the function that was at 2 is now at 6).
What is the general form combining shift, scale, reflection?
y = a·f(b(x−h)) + k.
In y=a·f(b(x−h))+k, define each parameter’s effect: a, b, h, k.
a=vertical scale ± reflection, b=horizontal scale ± reflection, h=horizontal shift, k=vertical shift.
Which transformations are done first, horizontal shifts or horizontal scalings?
Inside the parentheses: do scaling b then shift h. But practically we rewrite carefully to see the effect.
Which transformations are done after you handle the x-changes (like b(x−h))?
Outside transformations: vertical scaling by a, then vertical shift by k.
If we have y=−3f(2(x−1))+5, list transformations in order.
1) Shift right 1, 2) Horizontal compression by factor 1/2, 3) Reflect across x-axis, 4) Vertical stretch by 3, 5) Shift up 5.
Sometimes we combine reflection across y-axis if b<0 in f(bx). True?
Yes. If b<0, there’s also a reflection across the y-axis.
If a<0 in a·f(…), do we also have reflection across x-axis?
Yes, negative a includes reflection across x-axis plus scale factor |a|.
Give the domain shift for y=f(b(x−h)) if b>0.
Shift to x≥h if original domain was x≥0, then compress or stretch by factor 1/b along x.
For y=a·f(b(x−h))+k, how does the range shift if originally it was [m,∞) for f?
Scaled by factor a, reflected if a<0, then shifted up by k.
Explain how to rewrite y=a·f(b(x−h))+k if we want to see horizontal shift more clearly.
Inside: b(x−h) = b·x − b·h, so shift is h = (b·h)/b but we have to be mindful that dividing or factoring out b.
Define g(x)=−2f(3x+6). Which transformations are inside the parentheses?
3x+6 → factor out 3: 3(x+2). So horizontal shift left 2, then horizontal compression factor 1/3.
Continuing that example, what do −2 and +1 do?
−2 is reflection across x-axis plus vertical stretch by factor 2, +1 is shift up by 1.
If f(0)=2, what is g(0) for g(x)=2f(x)? Summarize the result.
g(0)=2×f(0)=2×2=4; vertical scale factor 2 at x=0.
In y = f(2x−4), we can factor out 2: y = f[2(x−2)]. Summarize transformations.
Shift right 2, then horizontal compression factor 1/2.
If we see y = f(−(x−3)), that is f(−x+3). Summarize transformations.
Shift right 3, reflect across y-axis, depending on sign inside. Actually it’s f(3−x), so it’s shift left 3? Carefully analyze.
Wait, f(−(x−3))=f(−x+3). That means x is replaced by (3−x). So is that a reflection across x=1.5 or simpler?
Yes, rewriting might help. But typically we see it as reflection across y-axis if we fix the expression. We must be precise.
Key tip: always isolate (x−h) by factoring out b. Then interpret sign for reflection. Good idea?
Yes. Factoring out the coefficient of x is the best approach.
If the inside coefficient of x is negative, we get reflection across the y-axis plus possible shift. T/F?
True.
If the outside coefficient is negative, we get reflection across the x-axis. T/F?
True.
In practice, is it best to handle horizontal transformations first or vertical transformations first?
Typically handle inside (horizontal) transformations first, then the outside (vertical).
When we do a horizontal shift x→(x−h), how does domain shift?
All x-values shift by +h. If domain was D, new domain is {x | x−h ∈ D}.
When we do a vertical shift up k, how does range shift?
All y-values shift up by k. If range was R, new range is R+k.
When we reflect across x-axis, how does range change if originally it was [m,∞)?
New range becomes (−∞, −m].
When we reflect across y-axis, does range necessarily change from the original?
No, reflection across y-axis affects domain (x) not range (y).
A horizontal scale factor by 1/2 (i.e. y=f(2x)) might do what to domain if the original domain was x≥0?
Now x≥0 is mapped to x≥0 still, but features occur twice as fast.
Vertical stretch by factor a>1 modifies range how?
All outputs are multiplied by a, so if range was [m,n], new range is [am, an] (assuming no reflection).
If a<0 in a vertical scale, e.g. −3, then there’s also reflection. True?
Yes, so the range flips sign plus scaling by 3.
If domain was (−∞, 5] for f(x), and we shift right 2, what’s new domain for g(x)=f(x−2)?
(−∞, 7].
If the range was all real y≥−1, after up shift by 3, it becomes y≥ what?
y≥2.
If the range was all real y≥1, after reflection across x-axis, it becomes y≤ −1. T/F?
True.
What is the recommended step to handle y = 2f(−(x+1)) − 3 first?
Rewrite inside: −(x+1)=−x−1, factor out negative, etc. Then interpret reflection, shift, scale step by step.
In transformations, is the order of horizontal transformations critical?
Yes, the order in which we apply shift vs. scale matters inside the argument.
Which is typically done first in y=f(b(x−h)) if we read left to right in the argument?
First shift x→(x−h), then scale x→b(x−h).
Can we do a combined transformation as x→b(x−h) in one step if we’re comfortable with factoring b out?
Yes, but carefully to avoid sign mistakes.
If we see y=−f(2(x+1))+4, list transformations in a clear sequence.
1) Shift left 1, 2) Horizontal compression by factor 1/2, 3) Reflect across x-axis, 4) Vertical shift up 4.
If the function is y=x² originally, how do we get y=−(x−3)²+2 in transformations?
Shift right 3, reflect across x-axis, shift up 2.
If we want to compress horizontally by factor 1/3 and shift up 5 from y=x³, how do we write that?
y = x³ becomes y=f(3x)+5 or y=(3x)³+5 = 27x³+5 if f(x)=x³.
Are transformations generally commutative (i.e. does shifting then reflecting yield same as reflecting then shifting)?
Not always, we must follow the correct order or rewrite carefully.
If we do y=|2x| vs. y=2|x|, which is a horizontal vs vertical change?
y=|2x| is horizontal compression by 1/2. y=2|x| is vertical stretch by 2.
List the transformations in y=2f(3x−6)−4, in correct order.
1) Shift right 2 (since 3(x−2)), 2) Horizontal compression factor 1/3, 3) Vertical stretch factor 2, 4) Shift down 4.
When f is piecewise, do transformations apply piecewise or overall?
They apply to each piece, adjusting intervals accordingly.
If a function is periodic (like sine/cosine), does horizontal scaling change its period?
Yes. y=f(bx) changes period to original period / b.
If f is odd, reflection across y-axis vs x-axis – do they produce the same or different shapes?
For an odd function, reflection across y-axis is the negative. But it’s not necessarily the same shape as reflection across x-axis. We must be cautious.
‘To compress horizontally by factor k’ means points get closer by factor k or 1/k?
If the formula is y=f(kx), actual numerical factor is 1/k. So if k>1, that’s a compression.
Why do we factor out the b from x−h to interpret horizontal transformations?
Because y=f(b(x−h)) is y=f[b(x−(h))], shifting is h units, not h/b. Factoring clarifies.
In a function transformation, can reflection be combined with scaling into a single factor a < 0?
Yes, a negative factor handles reflection + scale.
Are vertical transformations done after horizontal ones in standard function transformation approaches?
Yes. Typically we handle inside function transformations (horizontal) first, then outside (vertical).
If a function has domain [0,∞), after y=f(x−2), the domain is [2,∞). Why?
Because x≥2 ensures x−2≥0 for the new function.
If a function has range (−∞,10], and we do y=f(x)+5, new range is (−∞, 15]? T/F?
True, everything shifts up by 5.
Final tip: to identify transformations, always rewrite y in the form y=a·f(b(x−h))+k. T/F?
True. That clarifies each parameter’s effect.
Find f(3) if f(x) = 2x + 1.
f(3) = 2(3) + 1 = 7.
If g(x) = x², compute g(−2).
g(−2) = (−2)² = 4.
Let h(x) = 5 − x. Evaluate h(7).
h(7) = 5 − 7 = −2.
Find f(0) if f(x) = 3x² + 4.
f(0) = 3(0)² + 4 = 4.
If j(x) = x + 1, what is j(j(2))?
First j(2)=2+1=3; then j(3)=3+1=4.
For k(x)=|x|, evaluate k(−5).
|−5|=5.
If p(x)=x−3, solve p(x)=0.
x−3=0 → x=3.
Given r(x)=4−2x, find r(2) + r(−2).
r(2)=4−2(2)=0; r(−2)=4−2(−2)=4+4=8; sum=0+8=8.
If m(x)=2x+5, compute m(4)−m(1).
m(4)=2(4)+5=13; m(1)=2(1)+5=7; difference=13−7=6.
For f(x)=x²−1, find f(2)+f(−2).
f(2)=2²−1=4−1=3; f(−2)=4−1=3; sum=3+3=6.
If f(x)=x+2, for which x does f(x)=0?
x+2=0 → x=−2.
If g(x)=2x−3, solve g(x)=1.
2x−3=1 → 2x=4 → x=2.
Evaluate f(−1) if f(x)=3−x².
f(−1)=3−(−1)²=3−1=2.
Find h(2) if h(x)=3x+5 for x≥2. No domain issue, so h(2)=?
h(2)=3(2)+5=6+5=11.
If j(x)=−x, compute j(5).
j(5)=−5.
If k(x)=2, a constant function, find k(10).
Always 2.
For r(x)=x³, find r(−3).
r(−3)=−27.
Given p(x)=4−x, find p(0).
p(0)=4−0=4.
Compute q(1)+q(2) if q(x)=2x.
q(1)=2; q(2)=4; sum=6.
If f(x)=5x−1, for which x is f(x)=9?
5x−1=9 → 5x=10 → x=2.
If f(x)=x², compute f(3)−f(2).
f(3)=9; f(2)=4; difference=5.
Find domain & range for g(x)=|x| if no restrictions otherwise.
Domain: all real x. Range: [0,∞).
Evaluate h(1)+h(−1) if h(x)=x²+1.
h(1)=1+1=2; h(−1)=1+1=2; sum=4.
For j(x)=4−x², find j(0).
j(0)=4−0=4.
If m(x)=3x−4, find m(3) and m(−1).
m(3)=3(3)−4=9−4=5; m(−1)=3(−1)−4=−3−4=−7.
Compute p(2) for p(x)=2x+3.
p(2)=2(2)+3=7.
If f(x)=x−2, find x if f(x)=5.
x−2=5 → x=7.
For r(x)=−2x, evaluate r(−4).
r(−4)=−2(−4)=8.
If s(x)=x²+1, find s(1) & s(−1).
s(1)=2; s(−1)=2.
If f(x)=2x+5, find f(0)+f(−2).
f(0)=5; f(−2)=2(−2)+5=1; sum=6.
Let h(x)=2−x. For x=4, h(4)=?
2−4=−2.
If k(x)=3−2x, solve k(x)=0.
3−2x=0 → 2x=3 → x=1.5.
If j(x)=x+1, find j(0).
1.
Compute f(1)+f(2)+f(3) if f(x)=x.
1+2+3=6.
If g(x)=x²−4, find g(2).
2²−4=4−4=0.
For p(x)=−x, compute p(5). Then sum p(5)+p(−5).
p(5)=−5; p(−5)=5; sum=0.
If q(x)=1−2x, find x if q(x)=−3.
1−2x=−3 → −2x=−4 → x=2.
Let f(x)=x². Evaluate f(−1)+f(−2).
f(−1)=1; f(−2)=4; sum=5.
For r(x)=2x−1, find r(1), r(2), r(3).
r(1)=1; r(2)=3; r(3)=5.
If t(x)=|x−3|, evaluate t(3).
|3−3|=0.
If u(x)=x²−1, find u(1).
1²−1=0.
If h(x)=|x|−2, compute h(−2).
|−2|−2=2−2=0.
Let k(x)=3x. Solve k(x)=9.
3x=9 → x=3.
For a constant function c(x)=7, find c(−100).
7.
If g(x)=x−4, evaluate g(5)+g(−1).
g(5)=1; g(−1)=−5; sum=−4.
If f(x)=2−x², find f(2).
2−(2²)=2−4=−2.
For p(x)=4|x|, compute p(−3).
4|−3|=12.
If q(x)=x+2, find x for which q(x)=0.
x+2=0 → x=−2.
Compute r(2)+r(3) for r(x)=2x−5.
r(2)=−1; r(3)=1; sum=0.
If t(x)=−x+10, solve t(x)=4.
−x+10=4 → −x=−6 → x=6.
