Unit 5 · FunctionsF4 · Transformations · Guidance

Inverse Functions

Reverse a function, restrict domains when needed, and distinguish inverse from reciprocal.

30 min
Unlocks: Course Correction
Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

Finish the previous step to continue, or use “Show everything” above to explore.

Key terms

inverse function: a rule that swaps inputs and outputs

Step 2 of 7Try it firstDone

Try it first

Finish the previous step to continue, or use “Show everything” above to explore.

Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

Finish the previous step to continue, or use “Show everything” above to explore.

The decision that matters

Write $y=f(x)$ , swap $x$ and $y$ , then solve for $y$ . Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Write

y=f(x)

, swap

x

and

y

, then solve for

y

. Keep exact values through the setup; round only when the stem asks.

Core relationship

$f(f^{-1}(x))=x$ on the appropriate domain.

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Solve for the inverse algebraically to keep its domain and range clear. Graphing a function, its proposed inverse, and $y=x$ checks whether the two curves reflect across that line.

Explore in Desmos

Reflect a function across $y=x$

Graph y=f(x), its inverse, and y=x; inverse graphs reflect across y=x.

The calculator is live: change anything and see what happens.

Reflect a function across y=x

Degree mode

Loading Desmos…

Trap

The main trap

The notation

f^{-1}

means inverse function, not

1/f

Step 4 of 7Worked examplesDone

Worked examples

Finish the previous step to continue, or use “Show everything” above to explore.

Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

Find the inverse of

f(x)=3x-8

Write

y=3x-8

and swap:

x=3y-8

After swapping $x$ and $y$ , which equation must be solved for $y$ ?

f^{-1}(x)=\frac{x+8}{3}

Why does swapping input and output produce the inverse?

Worked example

Example 2: ACT twist

Problem

Find the inverse of

g(x)=(x-2)^2

when its domain is restricted to

x\ge2

Swap:

x=(y-2)^2

The restriction selects one square-root branch.

Which inverse formula respects outputs $y\ge2$ ?

g^{-1}(x)=2+\sqrt{x}

Why is the negative branch excluded?

Faded example · you finish it

Example 3: You finish it

Problem

Find

h^{-1}(10)

h(x)=2x+4

Solve

2x+4=10

Complete the calculation.

Desmos way

Check inverse symmetry

Graph y=f(x), its inverse, and y=x; inverse graphs reflect across y=x.

Check inverse symmetry

Degree mode

Loading Desmos…

Step 5 of 7PracticeDone

Practice

Finish the previous step to continue, or use “Show everything” above to explore.

Step 6 of 7Spot the mistakeDone

Spot the mistake

Finish the previous step to continue, or use “Show everything” above to explore.

Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Step 7 of 7Exit ticketDone

Exit ticket

Finish the previous step to continue, or use “Show everything” above to explore.

Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.

The decision that matters

Structure before arithmetic

Algebra or Desmos?

Reflect a function across y=xy=xy=x

The main trap

Check inverse symmetry

Reflect a function across $y=x$