Basic Function Types

The constant function f$x$ = b always outputs the same value regardless of input. It creates a horizontal line on a graph with domain (-∞,∞) and range that's just the single value {b}.

The identity function f$x$ = x returns exactly what you put in. Its graph is a straight line through the origin with domain and range both (-∞,∞). This simple function is surprisingly useful in many mathematical contexts.

Power functions show what happens when you raise x to different powers. The square function f$x$ = x² creates a U-shaped curve called a parabola with domain (-∞,∞) and range [0,∞). Notice that this function never outputs negative values! Meanwhile, the cubic function f$x$ = x³ has a distinctive S-shape with domain and range both (-∞,∞).

Try This! Graph f$x$ = x² and f$x$ = -x² on the same coordinate plane. How does the negative coefficient change the parabola's direction?

The square root function f$x$ = √x starts at the origin and curves upward, with domain [0,∞) and range [0,∞). This function is the inverse of x² for positive values, meaning it "undoes" the squaring operation.

More Function Building Blocks

The cube root function f$x$ = ∛x lets you work with roots beyond square roots. Unlike the square root function, it accepts negative inputs and produces negative outputs for them. Both its domain and range are (-∞,∞).

The reciprocal function f$x$ = 1/x creates hyperbolas that never cross the axes. Since division by zero is undefined, x can't equal zero, giving this function a domain of $-∞,0$∪(0,∞). Its range is also $-∞,0$∪(0,∞).

The squared reciprocal function f$x$ = 1/x² combines the reciprocal and squaring operations. Like the regular reciprocal function, it's undefined at x = 0, but since all outputs are squared, its range is [0,∞).

Important! Functions with asymptotes (like reciprocal functions) approach but never touch certain lines. This creates distinctive graph behaviors that are crucial for understanding limits in calculus.

The absolute value function f$x$ = |x| outputs the distance from zero on the number line. It creates a V-shaped graph with domain (-∞,∞) and range [0,∞). The function turns negative inputs into positive values while leaving positive inputs unchanged.

Exponential functions like f$x$ = aˣ or f$x$ = eᵏˣ grow dramatically with positive inputs. When a > 1 or k > 0, they increase rapidly; when 0 < a < 1 or k < 0, they decrease toward zero. Their domain is (-∞,∞) and range is (0,∞).

Logarithmic and Trigonometric Functions

The natural logarithmic function f$x$ = ln x is the inverse of eˣ. It grows very slowly as x increases, with domain (0,∞) and range (-∞,∞). This function is essential for solving exponential equations and modeling phenomena with diminishing returns.

The sine function f$x$ = sin x creates a smooth wave that repeats every 2π units. This periodic function has a domain of (-∞,∞) and range restricted to $-1,1$. Sine is fundamental for modeling oscillations and cycles in nature.

Similarly, the cosine function f$x$ = cos x also creates a wave pattern that repeats every 2π units. It's identical to sine but shifted horizontally, with domain (-∞,∞) and range $-1,1$. Cosine and sine work together to describe circular motion.

Visualization Tip: The sine and cosine functions can be viewed as tracking the y-coordinate and x-coordinate (respectively) of a point moving around the unit circle!

The tangent function f$x$ = tan x is defined as sin x/cos x. It has vertical asymptotes wherever cos x = 0 (at x = π/2 + kπ where k is any integer). Its domain excludes these points, and its range is all real numbers (-∞,∞).

The secant function f$x$ = sec x equals 1/cos x. Like tangent, it has vertical asymptotes where cos x = 0. Its range is $-∞,-1]∪[1,∞$, meaning it never outputs values between -1 and 1.

Inverse Trigonometric Functions

The inverse sine function f$x$ = sin⁻¹x (or arcsin x) reverses what the sine function does. Since sine outputs values only between -1 and 1, the inverse sine function only accepts inputs in this range. Its output is the angle (in radians) whose sine equals the input, with range $-π/2, π/2$.

The inverse cosine function f$x$ = cos⁻¹x (or arccos x) works similarly but finds the angle whose cosine equals the input. Its domain is $-1,1$ and its range is [0,π]. This more restricted range compared to inverse sine helps avoid ambiguity since cosine is even.

Remember: While sin(sin⁻¹x) always equals x, sin⁻¹(sin x) doesn't always equal x—it only works for angles between -π/2 and π/2!

The inverse tangent function f$x$ = tan⁻¹x (or arctan x) finds the angle whose tangent equals the input. Unlike the other inverse trig functions, it accepts any real number as input (domain is (-∞,∞)). However, its range is limited to $-π/2, π/2$, representing just one period of tangent's repeating pattern.

The relationship between inverse trig functions creates useful identities. For example, sec⁻¹x = cos⁻¹$1/x$, connecting the inverse secant directly to the inverse cosine function.