Composing & Inverting Functions — Challenge
Functions can be bolted together end to end, and any chain can be run backwards.
What these problems 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
If f(x)=x+1 and g(x)=2x, what is (f+g)(3)?
- 7
- 9
- 10
- 12
- 4
Worked answer
10
The sum of two functions is evaluated by evaluating each and adding.
f(3) = 4 and g(3) = 6.
4+6 = 10.
Problem 2
If f(x)=x2 and g(x)=x-3, what is f(g(5))?
- 4
- 22
- 25
- 49
- 2
Worked answer
4
Work from the inside out: g(5) = 5-3 = 2.
Then f(2) = 22 = 4.
The other order gives g(f(5)) = g(25) = 22, which is why 22 appears among the options.
Problem 3
Let f(x)=2x+3 and g(x)=x2-1. What is f(g(2))?
- 5
- 7
- 9
- 11
- 13
Worked answer
9
g(2)=22-1=3. Then f(3)=2(3)+3=9. The inner function runs first, which is why f(g(2)) is not g(f(2))=48.
Problem 4
Let f(x)=x+2 and g(x)=3x. For how many real x does f(g(x))=g(f(x))?
- 0
- 1
- 2
- Infinitely many
- Exactly 3
Worked answer
0
f(g(x))=3x+2 and g(f(x))=3(x+2)=3x+6. Setting them equal gives 3x+2=3x+6, so 2=6. That is never true, so there is no such x. Order genuinely matters.
Problem 5
A linear function satisfies f(1)=5 and f(3)=11. What is f-1(8)?
- 1
- 2
- 3
- 4
- 5
Worked answer
2
The slope is 11-53-1=3, so f(x)=3x+b. From f(1)=5: 3+b=5, so b=2 and f(x)=3x+2. f-1(8) asks which input gives 8: 3x+2=8 gives x=2.
Problem 6
For f(x)=x2 and g(x)=x+1, what is f(g(x))-g(f(x))?
- 0
- 1
- 2x
- 2x+1
- x2
Worked answer
2x
f(g(x))=(x+1)2=x2+2x+1. g(f(x))=x2+1. Subtracting: (x2+2x+1)-(x2+1)=2x. The two compositions differ everywhere except x=0.
Problem 7
If g(x)=2x-1 and f(g(x))=6x-1, what is f(x)?
- 3x+2
- 3x-2
- 6x+2
- 3x+1
- 2x+3
Worked answer
3x+2
Let u=2x-1, so x=u+12. Then f(u)=6 · u+12-1=3(u+1)-1=3u+2. So f(x)=3x+2. Check: f(2x-1)=3(2x-1)+2=6x-1.
Problem 8
Which function is its own inverse, so that f(f(x))=x for every x in its domain?
- f(x)=x+1
- f(x)=2x
- f(x)=1x
- f(x)=x2
- f(x)=x+2
Worked answer
f(x)=1x
Test f(x)=1x: f(f(x))=11/x=x. It undoes itself. Adding 1 twice gives x+2, and doubling twice gives 4x, so neither works. x2 is not one-to-one, so it has no inverse function at all.