A Function and Its Inverse on the Graph

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

An inverse exchanges input and output coordinates. Reflecting a graph across the line $y=x$ performs exactly this exchange.

Method

If $(a,b)$ lies on a bijective function's graph, $(b,a)$ lies on its inverse graph. Vertical and horizontal features exchange: domain becomes range, and a vertical asymptote becomes a horizontal one where applicable.

Worked example

The exponential $y=2^x$ passes through $(0,1)$ and $(1,2)$ . Its inverse $y=\log_2 x$ passes through $(1,0)$ and $(2,1)$ . The exponential's horizontal asymptote $y=0$ corresponds to the logarithm's vertical asymptote $x=0$ .

Common mistake

A graph and its inverse need not meet on the displayed window. Reflection does not mean reflection across the vertical axis.

Check your understanding

If $f(3)=8$ and $f$ is invertible, what is $f^{-1}(8)$ ?

Show answer

$3$ .

Explore

Exponential and logarithm

Try this. Compare bases 2 and 0.5: growth becomes decay. Move the input to 1; the logarithm is zero for either base.

BasePositive input x0.5

Base 2 · Blue: y = 2ˣ · Green: y = log_b(x) · Dashed: y = x. At x = 0.5, y = -1.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

If (2,7) lies on an invertible function’s graph, what lies on its inverse’s graph?

(2,1/7)

(7,2)

(−2,−7)

Hint 1 · Find a starting point

The inverse swaps the roles of input and output.

Hint 2 · Take the next step

Reflect the point across y=x.

Show the reasoning

Answer: (7,2)

f(2)=7 becomes f⁻¹(7)=2.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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