AP®MathsCollege BoardPrecalculusStudy GuidesPolynomial & Rational FunctionsTransformations of FunctionsDomain & Range of Transformed Functions

Domain & Range of Transformed Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Domain & range of transformed functions

How do transformations affect the domain and range of a function?

When a function is transformed
- the domain and range of the resulting function
- may be different from those of the original (parent) function
The effect depends on
- which type of transformation is applied
- and whether the domain and range of the original function are
  - bounded (have finite endpoints)
  - or unbounded (extend to $\infty$ or $- \infty$ )

How do vertical translations affect domain and range?

The transformation $g (x) = f (x) + k$ shifts the graph vertically by $k$ units
- The domain is unchanged
  - the inputs to the function have not been altered
- The range is shifted by $k$
  - If $f$ has range $[r, s]$ , then $g$ has range $[r + k, s + k]$
- An unbounded range remains unbounded, although a finite endpoint may be shifted
  - E.g., if $f$ has range $(- \infty, \infty)$ , then $g$ also has range $(- \infty, \infty)$
  - But if, for example, $f$ has range $[r, \infty)$ , then $g$ has range $[r + k, \infty)$

How do horizontal translations affect domain and range?

The transformation $g (x) = f (x + h)$ shifts the graph horizontally by $- h$ units
- The domain is shifted by $- h$
  - If $f$ has domain $[p, q]$ , then $g$ has domain $[p - h, q - h]$
- An unbounded domain remains unbounded, although a finite endpoint may be shifted
  - E.g., if the domain of $f$ is all real numbers (unbounded in both directions), then the domain of $g$ is also all real numbers
  - But if, for example, $f$ has domain $[p, \infty)$ , then $g$ has domain $[p - h, \infty)$
- The range is unchanged
  - the outputs of the function have not been altered

How do vertical dilations affect domain and range?

The transformation $g (x) = a f (x)$ stretches or compresses the graph vertically by a factor of $| a |$ , and reflects over the $x$ -axis if $a < 0$
- The domain is unchanged
  - the inputs to the function have not been altered
- The range is scaled by the factor $a$
  - If $a > 0$ and $f$ has range $[r, s]$ , then $g$ has range $[a r, a s]$
  - If $a < 0$ and $f$ has range $[r, s]$ , then $g$ has range $[a s, a r]$
    - the bounds are reversed because multiplying by a negative number flips the inequality
- An unbounded range remains unbounded, although a finite endpoint may be affected
  - E.g., if $f$ has range $(- \infty, \infty)$ , then $g$ also has range $(- \infty, \infty)$
  - But if, for example, $f$ has range $[0, \infty)$ , then a negative dilation flips it to $(- \infty, a \cdot 0] = (- \infty, 0]$

How do horizontal dilations affect domain and range?

The transformation $g (x) = f (b x)$ stretches or compresses the graph horizontally by a factor of $\frac{1}{| b |}$ , and reflects over the $y$ -axis if $b < 0$
- The domain is scaled by the factor $\frac{1}{b}$
  - If $b > 0$ and $f$ has domain $[p, q]$ , then $g$ has domain $[\frac{p}{b}, \frac{q}{b}]$
  - If $b < 0$ and $f$ has domain $[p, q]$ , then $g$ has domain $[\frac{q}{b}, \frac{p}{b}]$
    - the bounds are reversed because multiplying by a negative number flips the inequality
- An unbounded domain remains unbounded, although a finite endpoint may be affected
  - E.g., if the domain of $f$ is all real numbers (unbounded in both directions), then the domain of $g$ is also all real numbers
  - But if, for example, $f$ has domain $[0, \infty)$ , then a negative dilation flips it to $(- \infty, \frac{0}{b}] = (- \infty, 0]$
- The range is unchanged

What about combined transformations?

When multiple transformations are combined (e.g. $g (x) = a \cdot f (b (x + h)) + k$ ), apply the effects in sequence
- Horizontal transformations (translations and dilations) affect the domain
- Vertical transformations (translations and dilations) affect the range
Each transformation modifies the domain or range independently, so you can track them separately

When do transformations have no effect on domain or range?

If the domain of the parent function is all real numbers (unbounded in both directions)
- then horizontal translations and horizontal dilations leave the domain unchanged
- E.g. polynomial functions have domain all real numbers
  - any horizontal transformation still gives domain all real numbers
Similarly, if the range of the parent function is all real numbers (unbounded in both directions)
- then vertical translations and vertical dilations leave the range unchanged
- E.g. odd-degree polynomial functions have range all real numbers
  - any vertical transformation still gives range all real numbers
Transformations are most important for functions with restricted (bounded) domains or ranges
- E.g. $f (x) = \sqrt{x}$ , with domain $[0, \infty)$ and range $[0, \infty)$
- or piecewise-defined functions with specified domain intervals
Note that a rational function may have an unbounded domain but with specific values excluded which would make the denominator zero
- Although the overall domain remains unbounded under horizontal transformations
  - the excluded values are affected
- E.g. if $f (x) = \frac{1}{x - 2}$ has domain all real numbers where $x \neq 2$ ,
  - then $g (x) = f (x - 3) = \frac{1}{(x - 3) - 2} = \frac{1}{x - 5}$ has domain all real numbers where $x \neq 5$
  - The excluded value has shifted from $x = 2$ to $x = 5$ , matching the horizontal translation of $3$ units to the right

Worked Example

The function $f$ has domain $[- 2, 6]$ and range $[1, 9]$ .

The function $g$ is defined by $g (x) = 2 f (x - 4) + 3$ .

Find the domain and range of $g$ .

Answer:

Identify the transformations applied to $f$ to obtain $g$

$f (x - 4)$ : horizontal translation
- shift right by $4$ units (this affects the domain)
$2 f (\dots)$ : vertical dilation
- stretch by a factor of $2$ (this affects the range)
$\dots + 3$ : vertical translation
- shift up by $3$ units (this affects the range)

For the domain (affected by horizontal transformations only)

Start with the domain of $f$ , $[- 2, 6]$
The horizontal translation $f (x - 4)$ shifts the domain right by $4$

$[- 2 + 4, 6 + 4] = [2, 10]$

Domain of $g$ : $[2, 10]$

For the range (affected by vertical transformations only)

Start with the range of $f$ , $[1, 9]$
The vertical dilation by $2$ scales the range

$[2 (1), 2 (9)] = [2, 18]$

The vertical translation $. . . + 3$ shifts the range up by $3$

$[2 + 3, 18 + 3] = [5, 21]$

Range of $g$ : $[5, 21]$

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent