# 1.6 | A Library of Parent Functions

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

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

$$f(x) = a x^2$$

• 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 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 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 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

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

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 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 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 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

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

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

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

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