Composing & Inverting Functions — Practice
Functions can be bolted together end to end, and any chain can be run backwards.
What these questions cover
These questions use 3 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Function — A rule pairing each input with exactly one output.
- One-to-one — No two inputs share an output. Only a one-to-one function has an inverse that is itself a function.
- Horizontal line test — If any horizontal line meets a graph twice, the function is not one-to-one.
Problem 1
For f(x)=2x+1 and g(x)=x-4, find (f+g)(3).
Worked answer
6
f(3)=2(3)+1=7
g(3)=3-4=-1
7+(-1)=6
Problem 2
For f(x)=x2 and g(x)=3x, find (f+g)(2).
Worked answer
10
f(2)=4
g(2)=6
4+6=10
Problem 3
For f(x)=x+5 and g(x)=x-2, find (fg)(4).
Worked answer
18
f(4)=9
g(4)=2
9 × 2 = 18
Problem 4
For f(x)=x+3 and g(x)=2x, find f(g(5)).
Worked answer
13
g(5)=10
f(10)=13
Problem 5
For f(x)=x2 and g(x)=x-1, find f(g(4)).
Worked answer
9
g(4)=3
f(3)=9
Problem 6
For f(x)=2x-1 and g(x)=x2+1, find (f ° g)(2).
Worked answer
9
g(2)=5
f(5)=2(5)-1=9
Problem 7
For f(x)=x+4 and g(x)=3x, find g(f(2)).
Worked answer
18
f(2)=6
g(6)=18
Note f(g(2))=10 — the order matters
Problem 8
For f(x)=x2 and g(x)=x+3, write f(g(x)) expanded.
Worked answer
x^2+6x+9
f(g(x))=(x+3)2
=x2+6x+9
Problem 9
For f(x)=3x-2 and g(x)=x+1, write (f ° g)(x) simplified.
Worked answer
3x+1
3(x+1)-2
=3x+3-2=3x+1
Problem 10
Find the inverse of f(x)=x+7.
Worked answer
x-7
y=x+7, swap: x=y+7
y=x-7
Problem 11
Find the inverse of f(x)=4x.
Worked answer
x/4
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Problem 12
Find the inverse of f(x)=2x-6.
Worked answer
(x+6)/2
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Problem 13
Which function has an inverse that is also a function?
- f(x)=x2
- f(x)=|x|
- f(x)=x3
- f(x)=x2+1
Worked answer
f(x)=x3
x2 and |x| both send two inputs to one output
x3 is one-to-one — it never repeats a value
So only x3 can be undone
Problem 14
If f(x)=5x-3, find f-1(12).
Worked answer
3
5x-3=12
5x=15
x=3
Problem 15
For f(x)=1x and g(x)=x-2, which x must be excluded from f(g(x))?
Worked answer
2
f(g(x))=1x-2
The bottom is zero when x=2
Problem 16
For f(x)=x2-3, find f(f(2)).
Worked answer
-2
f(2)=4-3=1
f(1)=1-3=-2