Algebra and Functions Flashcards
What is a function?
A relation that assigns exactly one output to each input from a given domain.
What is the domain of a function?
The set of all possible input values (x-values) for which the function is defined.
What is the range of a function?
The set of all possible output values (y-values) of a function.
What is a one-to-one function?
A function where every element of the range corresponds to exactly one element of the domain.
What is a many-to-one function?
A function where multiple elements in the domain map to the same element in the range.
How do you test if a graph is a function?
Use the vertical line test — a vertical line should intersect the graph at most once.
What is the composition of functions?
Combining two functions such that the output of one becomes the input of another, written as (f ∘ g)(x) = f(g(x)).
What is the inverse of a function?
A function that reverses the effect of the original, denoted f⁻¹(x), where f(f⁻¹(x)) = x.
How do you find the inverse of a function algebraically?
Replace f(x) with y, swap x and y, then solve for y.
What condition must a function meet to have an inverse?
It must be one-to-one.
What does modulus |x| represent?
The distance of x from zero; always non-negative.
What is the effect of y = |f(x)| on a graph?
Reflects all negative y-values above the x-axis.
What is the effect of y = f(|x|) on a graph?
Reflects the graph for x > 0 across the y-axis.
What is the definition of an even function?
A function satisfying f(-x) = f(x) for all x; symmetric about the y-axis.
What is the definition of an odd function?
A function satisfying f(-x) = -f(x) for all x; symmetric about the origin.
What is a transformation of the graph y = f(x) to y = f(x + a)?
A horizontal translation a units to the left.
What does y = f(x) + a represent?
A vertical translation a units upward.
What is the graphical effect of y = af(x)?
Vertical stretch by a factor of a from the x-axis.
What is the effect of y = f(ax) on a graph?
Horizontal stretch/compression by a factor of 1/a.
What is a piecewise (or composite) function?
A function defined by different expressions over different parts of the domain.
