IMAT Mathematics · Functions

Inverse Functions and Transforming Graphs

An inverse function undoes a function and swaps its domain and range; shifts, stretches and reflections move a known graph without changing its basic shape.

All Functions notes

Why this matters

The 2023 paper asked for the inverse of a logarithmic function together with the set of values on which that inverse is defined. Transformations have not been asked yet, but they are on the syllabus and let you sketch any of the standard graphs after a shift or stretch.

Concept 1 of 2: Inverse functions: undoing a function

f

turns 4 into 9, the inverse turns 9 back into 4. To find it, write

y = f(x)

and solve for

x

: you get the input from the output. Inputs and outputs swap roles, so the domain of the inverse is the range of the original.

Definition

The inverse $f^{-1}$ of a function $f$ satisfies $f^{-1}(f(x)) = x$ and $f(f^{-1}(y)) = y$ .

It exists only when $f$ is one-to-one: no two inputs give the same output (so $x^2$ on all real numbers has no inverse, but on $x \ge 0$ it does).
Method: write $y = f(x)$ , solve for $x$ in terms of $y$ . That expression is $f^{-1}(y)$ .
The domain of $f^{-1}$ is the range of $f$ , and the other way round.
The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y = x$ .
$e^x$ and $\ln x$ are inverses: $\ln(e^x) = x$ and $e^{\ln x} = x$ for $x > 0$ .

Inverse function

y = f(x) \quad\Rightarrow\quad x = f^{-1}(y)

Worked example

Let

f(x) = 3\ln x + 2

for

x > 0

. Find

f^{-1}

and state where it is defined.

Practice this conceptself-check · 4 quick reps

The inverse function is not the reciprocal

f^{-1}(x)

means the function that undoes

f

;

\dfrac{1}{f(x)}

is the reciprocal, a different thing. For

f(x) = 2x

f^{-1}(x) = x/2

but

1/f(x) = 1/(2x)

Give the inverse with its domain

An option can have the right formula and the wrong domain. The domain of

f^{-1}

is the range of

f

: work out which outputs

f

really produces before choosing.

Concept 2 of 2: Transformations of graphs: shifts, stretches and reflections

Changes outside the function act on the output, so they move the graph up or down or stretch it vertically, exactly as written. Changes inside the bracket act on the input, so they work backwards:

f(x - 3)

reaches each value 3 units later, which moves the graph right, not left.

Definition

Starting from the graph of $y = f(x)$ :

$y = f(x) + k$ : shift up by $k$ (down if $k < 0$ ).
$y = f(x - h)$ : shift right by $h$ (left if $h < 0$ ).
$y = a f(x)$ : vertical stretch by factor $a$ .
$y = f(bx)$ : horizontal stretch by factor $1/b$ (squeezed towards the y-axis if $b > 1$ ).
$y = -f(x)$ : reflection in the x-axis; $y = f(-x)$ : reflection in the y-axis.
Shifts move asymptotes and key points with them, so they change domains and ranges.

Stretch, then shift

y = a\, f(x - h) + k

$a$ vertical stretch factor (a reflection in the x-axis if negative)
$h$ shift to the right
$k$ shift upwards

Worked example

Describe how to obtain

y = 2(x - 3)^2 + 1

from

y = x^2

and give its vertex. Then find the domain of

y = \ln(x + 2)

and where it crosses the x-axis.

Practice this conceptself-check · 4 quick reps

f(x minus 3) moves the graph right, not left

A change inside the bracket works against its sign:

f(x - 3)

is the graph moved 3 units right,

f(x + 3)

is it moved 3 units left. Outside the bracket the sign means what it says:

f(x) - 3

is 3 units down.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Inverse functions: undoing a function
Inverse function
$y = f(x) \quad\Rightarrow\quad x = f^{-1}(y)$
Transformations of graphs: shifts, stretches and reflections
Stretch, then shift
$y = a\, f(x - h) + k$

Watch out for (3)

The inverse function is not the reciprocal
→ Inverse functions: undoing a function
Give the inverse with its domain
→ Inverse functions: undoing a function
f(x minus 3) moves the graph right, not left
→ Transformations of graphs: shifts, stretches and reflections

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