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

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 $g\left(x\right)=f\left(x\right)+k$ 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 $\left(-\infty , \infty \right)$, then $g$ also has range $\left(-\infty , \infty \right)$

    • 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 $g\left(x\right)=f(x+h)$ 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 $g\left(x\right)=af\left(x\right)$ stretches or compresses the graph vertically by a factor of $\left|a\right|$, 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 $[ar, as]$

    • If $a<0$ and $f$ has range $[r, s]$, then $g$ has range $[as, ar]$

      • 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 $\left(-\infty , \infty \right)$, then $g$ also has range $\left(-\infty , \infty \right)$

    • But if, for example, $f$ has range $[0, \infty )$, then a negative dilation flips it to $(-\infty , a·0]=(-\infty , 0]$

How do horizontal dilations affect domain and range?

• The transformation $g\left(x\right)=f\left(bx\right)$ stretches or compresses the graph horizontally by a factor of $\frac{1}{\left|b\right|}$, 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 $\left[\frac{p}{b}, \frac{q}{b}\right]$

    • If $b<0$ and $f$ has domain $[p, q]$, then $g$ has domain $\left[\frac{q}{b}, \frac{p}{b}\right]$

      • 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. $g\left(x\right)=a·f\left(b\right(x+h\left)\right)+k$), 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. $f\left(x\right)=\sqrt{x}$, 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 $f\left(x\right)=\frac{1}{x-2}$ has domain all real numbers where $x\ne 2$,

    • then $g\left(x\right)=f(x-3)=\frac{1}{(x-3)-2}=\frac{1}{x-5}$ has domain all real numbers where $x\ne 5$

    • The excluded value has shifted from $x=2$ to $x=5$, matching the horizontal translation of $3$ units to the right