Combining Functions Flashcards
Define as an equation the combination of both functions:
f + g
f(x) + g(x) = (f + g)(x)
Define as an equation the combination of both functions:
f - g
f(x) - g(x) = (f - g)(x)
Define as an equation the combination of both functions:
f * g
f(x) * g(x) = (f * g)(x)
Define as an equation the combination of both functions:
f/g
f(x)/g(x) = (f / g)(x)
Write mathematically: f composed with g of x.
Write mathematically: f composed with g of 2.
Write mathematically: h composed with g of 2.
Write mathematically: f composed with p of 6.
Write mathematically: f composed with g of h of 5.
f(g(h(5)))
Write mathematically: f composed with g of h of x.
Write mathematically: c composed with g of f of x.
Write mathematically: g composed with t of f of 6.
3
7
Which of the following best approximates the value of g(h(1))?
-7, -5, 0 or 2
-5
What is f of g?
What is g(f(x))?
g(f(x)) = 2^2+3
1.𝑓(𝑔(0)) = 26, 𝑔(𝑓(0)) = −57
2. 𝑓(𝑔(0)) = 27, 𝑔(𝑓(0))= -94
3.𝑓(𝑔(0))= 4, 𝑔(𝑓(0))= 4
4.𝑓(𝑔(0))= 1/5 ,𝑔(𝑓(0))= 5
- Find f composed in g and simplify.
- Find g composed in f and simplify.
- Find g composed with f
What is the domain of g composed with f?
[0,6]
Why should we find the domain of a function BEFORE any simplification?
Because simplification could change the whole function. For example:
What are the six graphs to ALWAYS remember?
What would be the formulas to shift a function up or down by c units?
Give an example of vertical shifting using f(x) = x²
What would be the formulas to shift a function left or right by c units?
Give an example of left/right shifting using f(x) = lxl.
What would be the formulas for compressing or stretching vertically a function by c units assuming c >1?
Give an example of compressing or stretching using f(x) = x³.
y = cf(x³) stretching vertically
y = (1/c)f(x³) compressing vertically
What would be the formulas for compressing or stretching horizontally a function by c units assuming c >1?
Give an example of compressing or stretching using f(x) = mx+b.
What would be the manipulation on the y = f(x) to cause a reflection over the x-axis?
Give an example of reflection over the x-axis using f(x) = x².
What would be the manipulation on the y = f(x) to cause a reflection over the y-axis?
Give an example of reflection over the y-axis using f(x) = √x.
f(x)= -√x.
Use transformations to sketch the following graph and describe the steps.
Use transformations to sketch the following graph and describe the steps.
Use transformations to sketch the following graph and describe the steps.
If f(x) is our initial function, how would we describe the new function g(x) = f(x) + k?
g(x) is a vertical shift of the function f(x), where all the output values have been increased by k.
In g(x) = f(x) + k:
If k is positive, then the graph will shift________
If k is negative, then the graph will shift _______
If k is positive, then the graph will shift up
If k is negative, then the graph will shift down
A function f(x) is given as a table below. Create a table for the function g(x) = f (x) - 3
If f(x) is our initial function, how would we describe the new function g(x) = f(x + k)?
k is a constant
then g(x) is a horizontal shift of the function f(x)
Let f(x) be our initial function and g(x) = f(x + k) be our transformation.
If k is positive, then the graph will shift ______
If k is negative, then the graph will shift _______
If k is positive, then the graph will shift left
If k is negative, then the graph will shift right
A function f(x) is given as a table below. Create a table for the function g(x) = f (x - 3):
Given f (x) = l x l , sketch a graph of h(x) = f (x + 1) - 3 and find a formula for the new h function.
Write a formula for this function:
h(x) = (√x -1) +2
Reflect the graph below vertically and write the appropriate formula for s(x).
s(x) = - √x
Reflect the graph below horizontally and write the appropriate formula for s(x).
s(x) = √ (-x)
Given a function f(x), if we define a new function g(x) as g(x) = _______________
then g(x) is a ___________reflection of the function f(x), sometimes called a reflection about the ________________.
g(x) = - f(x)
g(x) is a vertical reflection of the function f(x), sometimes called a reflection about the x-axis
If we define a new function g(x) as g(x) = f(- x),
then g(x) is a ______________reflection of the function f(x), sometimes called a reflection about the________.
g(x) = f(- x),
then g(x) is a horizontal reflection of the function f(x), sometimes called a reflection about the y-axis
A function f(x) is given as a table below. Create a table for the function g(x) = - f (x) and h(x) = f (- x).
Given a function f(x), if we define a new function g(x) as g(x) = kf (x) , where k is a constant
then g(x) is a ______________or _____________ of the function f(x).
Given a function f(x), if we define a new function g(x) as g(x) = kf (x) , where k is a constant
then g(x) is a vertical stretch or compression of the function f(x).
Given a function f(x), if we define a new function g(x) as g(x) = f (kx), where k is a constant
then g(x) is a ____________________or. ______________________ of function f(x).
Given a function f(x), if we define a new function g(x) as g(x) = f (kx), where k is a constant
then g(x) is a horizontal stretch or compression of the function f(x).
If k > 1, then the graph will be_________________
If 0< k < 1, then the graph will be _______________
If k < 0, then there will be ______________________________________
If k > 1, then the graph will be compressed by 1k
If 0< k < 1, then the graph will be stretched by 1k
If k < 0, then there will be a combination of a horizontal stretch or compression with a horizontal reflection.
A function f(x) is given in the table below. Create a table for the function g(x) = f (1/2*x)
The graph of 𝑦=𝑓(𝑥)
is given below:
On a piece of paper sketch the graph of 𝑦=𝑓(𝑥+2)
and determine the new coordinates of points A, B, and C.
A = (-3,0)
B = (0,1)
C = (3,0)
The graph of 𝑦=𝑓(𝑥)
is given below:
On a piece of paper sketch the graph of 𝑦=−𝑓(𝑥)+6
and determine the new coordinates of points A, B, and C.
A = (-1,6)
B = (2,5)
C = (5,6)
Let 𝑓(𝑥)= 𝑥³ + 1 and let 𝑔(𝑥)= 𝑥 + 1.
Match the functions defined below with the letters labeling their equivalent expressions.
Provide the expanded forms.
1. 𝑔(𝑥²)
2. (𝑔(𝑥))²
3. 𝑔(𝑥)𝑓(𝑥)
4. (𝑓(𝑥))²
- 1 + x²
- x² + 2x + 1
- 1 + x + x³ + x⁴
- 1 + 2x³ + x⁶
Find a formula for: g(f(h(x))) =
Let 𝑓(𝑥) = √(6−𝑥) and 𝑔(𝑥)= 𝑥² − 𝑥.
Then the domain of (𝑓∘ 𝑔) is equal to [𝑎,𝑏]
for a = _____ and b = _______.
Draw the graph.
[-2,3]
Find the inverse of the following functions:
Suppose f(x) = x + 4 and g(x) = 2x -5.
Then:
Consider the following function:
Let f(x) = √(5 - 4x)
Find 3 decompositions of 𝑓(𝑥)=𝑝(𝑞(𝑥)) into a pair of functions 𝑝(𝑥) (the outside function) and 𝑞(𝑥) (the inside function) making the composition true.
p(x) = ______ and q(x) =_________
p(x) = ______ and q(x) =_________
p(x) = ______ and q(x) =_________
Decompose the function below into 𝑢(𝑣(𝑥)). In each part, based on the function 𝑣(𝑥)
given, find the corresponding 𝑢(𝑥)
needed to decompose the function.
