A composition runs the output of one function into the input of another. f(g(x)) means do g first, then f.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Order matters
f(g(x)) and g(f(x)) are usually different functions. Putting on socks then shoes is not the same as shoes then socks.
An inverse undoes
The inverse of f sends every output back to the input it came from. Written f⁻¹, the −1 is a name, not a power.
Finding one
Swap x and y, then solve for y. The swap is what reverses the roles of input and output.
The check
A candidate is the inverse only if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Compose in both directions.
Not every function has one
If two inputs share an output, nothing can send that output back to both. y = x² needs its domain restricted before it has an inverse.
Composition feeds one into the other
f(g(x)) means apply g first, then f. The notation reads right to left, which catches people out. Writing out the substitution explicitly rather than trying to do it mentally avoids getting the order backwards.
The domain shrinks
For f(g(x)) to make sense, x must be in the domain of g and g(x) must be in the domain of f. Composition often restricts the domain further than either function alone, and that restriction must be stated.
An inverse undoes the function
f⁻¹ takes the output back to the input, so f(f⁻¹(x)) = x. To find it, swap x and y in the equation and solve for y. That swap is exactly the statement that inputs and outputs have exchanged roles.
Not every function has an inverse
If two inputs give the same output, the inverse would have to send one input to two — not a function. Only one-to-one functions are invertible, which is tested by the horizontal line test. Restricting the domain can rescue the rest.
Step 2: Try It Yourself
Tap and try it out.
The slope is 2: for every 1 across, the line goes 2 up.
Step 3: Watch an Example
One step at a time.
Watch Nadia Invert f(x) = 3x + 6
Nadia needs the function that undoes this one.
- Step 1
She writes it as y = 3x + 6.
Step 4: Your Turn
Practice makes it stick.
The Composition
Problem 1 of 2
f(x) = 2x and g(x) = x + 3. What is f(g(4))?
The Other Order
Problem 2 of 2
With the same functions, what is g(f(4))?
In and Out
1 of 8
f(x) = x + 5. What is f⁻¹(12)?
2 of 8
f(x) = 4x. What is f⁻¹(20)?
3 of 8
f(x) = 2x − 1 and g(x) = x². What is f(g(3))?
4 of 8
Same functions. What is g(f(3))?
5 of 8
f(x) = 3x + 6. What is f⁻¹(0)?
6 of 8
f(f⁻¹(9)) equals what?
7 of 8
Put the steps for finding an inverse in order.
- 1Swap x and y.
- 2Solve the new equation for y.
- 3Verify by composing in both directions.
- 4Write the function as y = something.
8 of 8
Which functions have an inverse over all real numbers?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x) = x − 7. What is f⁻¹(3)?
Question 2 of 2
Why does y = x² have no inverse over all real numbers?
What You Learned
- f(g(x)) means do g first, then f, and the order matters.
- To find an inverse, swap x and y, then solve for y.
- Verify by composing both ways; both must return x.