Functions

What a function is

A function gives exactly one output for each input. A graph is a function if it passes the vertical line test.

Example: \(\{(1, 2), (2, 3), (1, 5)\}\) is not a function: the input 1 has two outputs.

Function notation

Function notation: \(f(x)\) is the output when the input is \(x\). To find \(f(\text{something})\), replace every \(x\) with that something, in parentheses.

Example: \(f(x) = x^2 - 3x\): \(f(-2) = (-2)^2 - 3(-2) = 4 + 6 = 10\).

Example: \(f(a + 1) = (a + 1)^2 - 3(a + 1) = a^2 - a - 2\).

Watch out: \(f(x)\) is not "\(f\) times \(x\)".

Reading tables and graphs

\(f(3)\): find input 3, read the output (the \(y\)-value). \(f(x) = 4\): find every input whose output is 4.

Example: If the graph passes through \((3, -1)\), then \(f(3) = -1\).

Domain from a formula

The domain is every allowed input. Start with all real numbers, then remove: values that make a denominator 0, and values that make an even root negative.

Example: \(f(x) = \dfrac{1}{x - 5}\): \((-\infty, 5) \cup (5, \infty)\).

Example: \(g(x) = \sqrt{2x - 6}\): \(2x - 6 \ge 0\), so \([3, \infty)\).

Example: \(h(x) = \dfrac{x}{x^2 - 4}\): all real numbers except \(2\) and \(-2\). A polynomial: \((-\infty, \infty)\).

Domain and range from a graph

Domain: how far the graph reaches left to right (the \(x\)-values). Range: how far it reaches bottom to top (the \(y\)-values). Closed dot: [ ]. Open dot: ( ). Arrow: \(\infty\).

Example: \(f(x) = x^2 - 4\): domain \((-\infty, \infty)\), range \([-4, \infty)\). \(\;\sqrt{x}\): domain \([0, \infty)\), range \([0, \infty)\).

Basic graphs to know

Transformations

A transformation moves or reshapes a basic graph \(f\):

Example: \(y = -(x - 2)^2 + 3\): \(x^2\) moved right 2, flipped, up 3; vertex \((2, 3)\). \(\;y = \sqrt{x + 4} - 1\) starts at \((-4, -1)\).

Watch out: changes inside the parentheses go the opposite way: \(x - 2\) moves right.

Composition

Composition: \((f \circ g)(x) = f(g(x))\). Work inside out: find \(g\) first, then put that into \(f\). Order matters.

Example: \(f(x) = 2x + 1\), \(g(x) = x^2\): \(f(g(x)) = 2x^2 + 1\), but \(g(f(x)) = (2x + 1)^2 = 4x^2 + 4x + 1\).

Example: \((f \circ g)(3) = f(g(3)) = f(9) = 19\).

Inverse functions

Only a one-to-one function (passes the horizontal line test) has an inverse. To find \(f^{-1}\): write \(y = f(x)\), swap \(x\) and \(y\), solve for \(y\). Check: \(f(f^{-1}(x)) = x\). The graph of \(f^{-1}\) is \(f\) reflected over \(y = x\); domain and range swap.

Example: \(f(x) = 3x - 5\): \(x = 3y - 5\), so \(f^{-1}(x) = \dfrac{x + 5}{3}\). Check: \(3 \cdot \dfrac{x + 5}{3} - 5 = x\).

Example: \(f(x) = \sqrt[3]{x - 1}\): \(x = \sqrt[3]{y - 1}\), so \(f^{-1}(x) = x^3 + 1\).

Watch out: \(f^{-1}(x)\) is not \(\dfrac{1}{f(x)}\).