transformations of functions
Arithmetic Combinations of Functions
inverse functions
Modeling
Interpreting
100
Given
f(x)=x^2
, write the equation for the function shifted 3 units right and 2 units up.
What is
f(x)=(x-3)^2+2?
100
Given
f(x)=x^2-7x and g(x)=x+5
find
(f+g)(x)
What is
x^2-6x+5
?
100
find the inverse function of f.
f(x)=3x+1
What is
f^-1(x)=(x-1)/3
?
100
Given:
f(x)=2x+3 and g(x)=x^2-1
Find and simplify
(f+g)(x).
.
What is
(f+g)(x)=x^2+2x+2
?
100
Consider the function:
g(x)=-(-x+3)^2+5
Describe the transformations that were made to the parent function
f(x)=x^2
.
What is function
- Reflected over the x-axis
- Reflected over the y-axis
- Shifted 3 units left
- Shifted 5 units up
?
200
Given:
f(x)=x^2
The function is transformed to:
g(x)=(x-3)^2+2
Calculate
g(5)
.
What is 6?
200
Given
f(x)=x^2-7x and g(x)=x+5
find
(f-g)(x)
What is
x^2-8x-5
?
200
find the inverse function of f.
f(x)=x^3+1
What is
f^-1(x)=root(3)(x-1)
?
200
Given
f(x)=x^2-7x and g(x)=x+5
find
(gof)(x)
What is
x^2-7x+5
?
200
Given:
f(x)=x+4 and g(x)=2x
A student says:
(f\circ g)(x)=(g\circ f)(x)
because multiplication and addition can be done in either order.
Is the student's reasoning correct? Explain.
What if
(f\circ g)(x)=f(2x)=2x+4
while
(g\circ f)(x)=g(x+4)=2x+8
Therefore, the student's reasoning would be incorrect?
300
Use the graph of
f(x)=x^2
to write an equation for the function represented by each graph.
What is
f(x)=-(x+1)^2+1
?
300
Given
f(x)=x^2-7x and g(x)=x+5
find
(fg)(x)
What is
x^3-2x^2-35x
?
300
find the inverse function of f.
f(x)=(x^5)/4
What is
f^-1(x)=root(5)(4x)
?
300
Use the graph of
f(x)=sqrtx
to write an equation for the function represented by each graph.
What is
f(x)=sqrt(x+1)-7
?
300
Given:
f(x)=2x+5 and g(x)=3x
A student calculates:
(f\circ g)(x)=6x+5
Explain what
(f(g(x))
means.
Why is the output of
g(x)
used as the input of
f(x)
?
400
Describe how
f(x)=x^2
was transformed to
g(x)=-(x+2)^2-5
What is a reflection across the x-axis, horizontal shift 2 units to the left, and vertical shift down by 5 units?
400
Given
f(x)=x^2-7x and g(x)=x+5
find
(fog)(x)
What is
x^2+3x-10
?
400
verify that f and g are inverse functions.
f(x)=7x+1 and g(x)=(x-1)/7
What is
f(g(x))=7((x-1)/7)+1=x-1+1=x
g(f(x))=(7x+1-1)/7=(7x)/7=x
?
400
Parent function:
f(x)=x^2
The function is shifted 3 units to the right and 4 units down.
Write an equation for the transformed function g(x).
What is
g(x)=(x-3)^2-4
?
400
Given:
f(x)=3x+1
and
f^{-1}(x)=\frac{x-1}{3}
Explain what
f^{-1}(10)
represents in relation to
f(x).
Expected answer:
\(f^{-1}(10)\) tells us the input value that produces an output of 10 when using \(f(x)\).
Since:
\[ f^{-1}(10)=3 \]
we know:
\[ f(3)=10 \]
What if
f^{-1}(10)
tells us the input value that produces an output of 10 when using
f(x)
Since:
f^{-1}(10)=3
we know:
f(3)=10
500
What transformations are represented by
g(x)=(-x-3)^2+4
What is a reflection across the y-axis, a horizontal shift to the right by 3 units and a vertical shift up by 4 units?
500
Given
f(x)=x^2-7x and g(x)=x+5
find
(fog)(2)
What is
0?
500
verify that f and g are inverse functions.
f(x)=x^3+5 and g(x)=root(3)(x-5)
What is
f(g(x))=(root(3)(x-5))^3+5=x-5+5=x
g(f(x))=root(3)(x^3+5-5)=root(3)x^3-=x
?
500
A taxi company charges a $5 starting fee plus $2 per mile. The cost is modeled by:
C(m)=2m+5
Create a function
C^-1(x)
that determines the number of miles traveled when you know the total cost.
What is
C^-1(x)=(x-5)/2
?
500
A student says: "The inverse function gives you the opposite of the original function."
Is the student correct? Explain.
What if the inverse doesn't simply make the function's values "opposite." Instead, it reverses the input-output relationship:
f(a)=b \quad\leftrightarrow\quad f^{-1}(b)=a
?