1.6 | A Library of Parent Functions


Linear

Linear Function
$$
f(x) = mx + b
$$


  • Domain: \((-\infty, \infty)\)
  • Range: \((-\infty, \infty)\)
  • \(x\)-intercept: \((-b/m,0)\)
  • \(y\)-intercept: \((0,b)\)
  • Increasing when \(m > 0\)
  • Decreasing when \(m < 0\)
Absolute

Absolute Function
$$
f(x) = |x|
$$


  • Domain: \((-\infty, \infty)\)
  • Range: \([0, \infty)\)
  • \(x\)-intercept: \((0,0)\)
  • \(y\)-intercept: \((0,0)\)
  • Decreasing on \((-\infty,0)\)
  • Increasing on \((0,\infty)\)
  • \(y\)-axis symmetry
Square Root

Square Root Function
$$
f(x) = \sqrt{x}
$$


  • Domain: \([0, \infty)\)
  • Range: \([0, \infty)\)
  • \(x\)-intercept: \((0,0)\)
  • \(y\)-intercept: \((0,0)\)
  • Increasing on \((0,\infty)\)
Quadratic

Quadratic (Squaring) Function
$$
f(x) = a x^2
$$


  • Domain: \((-\infty, \infty)\)
  • Range \(a>0\): \([0, \infty)\)
  • Range \(a<0\): \((-\infty, 0]\)
  • \(x\)-intercept: \((0,0)\)
  • \(y\)-intercept: \((0,0)\)
  • Decreasing on \((-\infty,0)\) for \(a>0\)
  • Increasing on \((0,\infty)\) for \(a>0\)
  • Increasing on \((-\infty,0)\) for \(a<0\)
  • Decreasing on \((0,\infty)\) for \(a<0\)
  • Even function
  • \(y\)-axis symmetry
  • Relative minimum (\(a > 0\)), relative maximum (\(a < 0 \)), or vertex \((0,0)\)
Cubic

Cubic Function
$$
f(x) = x^3
$$


  • Domain: \((-\infty, \infty)\)
  • Range: \((-\infty, \infty)\)
  • \(x\)-intercept: \((0,0)\)
  • \(y\)-intercept: \((0,0)\)
  • Increasing on \((-\infty,\infty)\)
  • Odd function
  • Origin symmetry
Rational

Rational Function
$$
f(x) = \frac{1}{x}
$$


  • Domain: \((-\infty,0) \cup (0, \infty)\)
  • Range: \((-\infty,0) \cup (0, \infty)\)
  • Decreasing on \((-\infty,0) \cup (0, \infty)\)
  • Odd function
  • Origin symmetry

Exponential Function
$$
f(x) = a^x, \quad a > 1
$$


  • Domain: \((-\infty, \infty)\)
  • Range: \((0, \infty)\)
  • \(y\)-intercept: \((0,1)\)
  • Increasing on \((-\infty,\infty)\) for \(f(x) = a^{x}\)
  • Decreasing on \((-\infty,\infty)\) for \(f(x) = a^{-x}\)
  • Horizontal Asymptote: \(x\)-axis

Logarithmic Function
$$
f(x) = \log_a x, \; a > 0, \; a \not = 1
$$


  • Domain: \((0, \infty)\)
  • Range: \((-\infty, \infty)\)
  • \(x\)-intercept: \((1,0)\)
  • Increasing on \((0,\infty)\)
  • Vertical Asymptote: \(y\)-axis
Sine

Sine Function
$$
f(x) = \sin x
$$


  • Domain: \((-\infty, \infty)\)
  • Range: \([-1, 1]\)
  • Period: \(2 \pi\)
  • \(x\)-intercept: \((n \pi,0)\)
  • \(y\)-intercept: \((0,0)\)
  • Odd function
  • Origin Symmetry
Sine

Cosine Function
$$
f(x) = \cos x
$$


  • Domain: \((-\infty, \infty)\)
  • Range: \([-1, 1]\)
  • Period: \(2 \pi \)
  • \(x\)-intercept: \( \left( \frac{\pi}{2}+ n\pi,0 \right) \)
  • \(y\)-intercept: \((0,1)\)
  • Even function
  • \(y\)-axis Symmetry
Sine

Tangent Function
$$
f(x) = \tan x
$$


  • Domain: all \(x \not = \frac{\pi}{2} + n \pi \)
  • Range: \((-\infty, \infty)\)
  • Period: \( \pi \)
  • \(x\)-intercept: \( \left( n \pi,0 \right) \)
  • \(y\)-intercept: \((0,0)\)
  • Vertical Asymptotes: \(x = \frac{\pi}{2} + n \pi \)
  • Odd function
  • Origin Symmetry
Cosecant

Cosecant Function
$$
f(x) = \csc x
$$


  • Domain: all \(x \not = n \pi\)
  • Range: \((-\infty, -1] \cup [1, \infty)\)
  • Period: \(2 \pi\)
  • Vertical Asymptotes: \(x = n \pi\)
  • Odd function
  • Origin Symmetry
Secant

Secant Function
$$
f(x) = \sec x
$$


  • Domain: all \(x \not = \frac{\pi}{2}+n \pi\)
  • Range: \((-\infty, -1] \cup [1, \infty)\)
  • Period: \(2 \pi\)
  • Vertical Asymptotes: \(x = \frac{\pi}{2}+n \pi\)
  • Even function
  • \(y\)-axis Symmetry
Cotangent

Cotangent Function
$$
f(x) = \cot x
$$


  • Domain: all \(x \not = n \pi\)
  • Range: \((-\infty,\infty)\)
  • Period: \(\pi\)
  • \(x\)-intercepts: \(\left( \frac{\pi}{2} + n\pi, 0 \right) \)
  • Vertical Asymptotes: \(x = n \pi\)
  • Odd function
  • Origin Symmetry
Arcsine

Inverse Sine Function
$$
f(x) = \sin^{-1} x
$$


  • Domain: \([-1,1]\)
  • Range: \([-\pi/2,\pi/2]\)
  • \(x\)-intercept: \((0,0)\)
  • \(y\)-intercept: \((0,0)\)
  • Odd function
  • Origin Symmetry
Arccosine

Inverse Cosine Function
$$
f(x) = \cos^{-1} x
$$


  • Domain: \([-1,1]\)
  • Range: \([0,\pi]\)
  • \(x\)-intercept; \((1,0)\)
  • \(y\)-intercept: \((0,\pi/2)\)
Arctangent

Inverse Tangent Function
$$
f(x) = \tan^{-1} x
$$


  • Domain: \((-\infty,\infty)\)
  • Range: \((-\pi/2,\pi/2)\)
  • \(x\)-intercept: \((0,0)\)
  • \(y\)-intercept: \((0,0)\)
  • Horizontal Asymptotes: \(y = \pm \frac{\pi}{2} \)
  • Odd function
  • Origin Symmetry

E 1.6 Exercises