Reciprocal & Rational Functions (DP IB Analysis & Approaches (AA): SL): Revision Note

Reciprocal functions & graphs

What is the reciprocal function?

• The reciprocal function is defined by $f\left(x\right)=\frac{1}{x},x\ne 0$

• Its domain is the set of all real values except 0

• Its range is the set of all real values except 0

• The reciprocal function has a self-inverse nature

  • $f^{-1}\left(x\right)=f\left(x\right)$

  • $(f\circ f)\left(x\right)=x$

What are the key features of the reciprocal graph?

• The graph does not have a y-intercept

• The graph does not have any roots

• The graph has two asymptotes

  • A horizontal asymptote at the x-axis: $y=0$

    • This is the limiting value when the absolute value of x gets very large

  • A vertical asymptote at the y-axis: $x=0$

    • This is the value that causes the denominator to be zero

• The graph has two axes of symmetry

  • $y=x$

  • $y=-x$

• The graph does not have any minimum or maximum points

Linear rational functions & graphs

What is a rational function with linear expressions?

• A (linear) rational function is of the form $f\left(x\right)=\frac{ax+b}{cx+d},x\ne -\frac{d}{c}$

  • e.g. $f\left(x\right)=\frac{2x-3}{5x+1}$

• The reciprocal function is a special case of a rational function

• The inverse is also a rational function $f^{-1}\left(x\right)=\frac{-dx+b}{cx-a}$

  • You do not need to remember this formula

    • You can derive the inverse easily in your exam

What are the key features of linear rational graphs?

Intersections with coordinate axes

• The graph has a y-intercept at $(0,\frac{b}{d})$ provided $d\ne 0$

  • Substitute $x=0$ to find the y-coordinate

    • e.g. the y-intercept of $f\left(x\right)=\frac{2x-3}{5x+1}$ is $\left(0,-3\right)$

    • e.g. $f\left(x\right)=\frac{2x-3}{5x}$ does not have a y-intercept

• The graph has one root at $(-\frac{b}{a},0)$ provided $a\ne 0$

  • Set the numerator equal to zero and solve

    • e.g. the root of $f\left(x\right)=\frac{2x-3}{5x+1}$ is $\left(\frac{3}{2},0\right)$

    • e.g. $f\left(x\right)=\frac{-3}{5x+1}$ does not have any roots

Asymptotes

• The graph has two asymptotes

  • A horizontal asymptote: $y=\frac{a}{c}$

    • This is the limiting value when the absolute value of x gets very large

    • e.g. the horizontal asymptote of $f\left(x\right)=\frac{2x-3}{5x+1}$ is $y=\frac{2}{5}$

  • A vertical asymptote: $x=-\frac{d}{c}$

    • This is the value that causes the denominator to be zero

    • e.g. the horizontal asymptote of $f\left(x\right)=\frac{2x-3}{5x+1}$ is $x=-\frac{1}{5}$

Domain and range

• Its domain is the set of all real values except  $-\frac{d}{c}$

  • It is undefined for the value of $x$ which causes the denominator to equal zero

    • e.g. the domain of $f\left(x\right)=\frac{2x-3}{5x+1}$ is $x\ne -\frac{1}{5}$

• Its range is the set of all real values except $\frac{a}{c}$

  • e.g. the range of $f\left(x\right)=\frac{2x-3}{5x+1}$ is $f\left(x\right)\ne \frac{2}{5}$

Turning points

• The graph does not have any minimum or maximum points