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 \((0,\infty)\)

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 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