The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 21 (Intermediate Algebra) Functions and Function Operations

Overview

This lecture opens the chapter on functions: what a function is, how to recognize one, and how to combine functions to build new ones. A function is a rule that assigns to each input exactly one output, and we test that "one output per input" condition either graphically (the vertical line test) or algebraically (solve for $y$ and check for a $\pm$). From there we look at the domain of a function — the set of legal inputs — guarding against the two operations that break a real-valued function: dividing by zero and taking the square root of a negative. Finally we learn function evaluation (substitution), the five function operations (sum, difference, product, quotient, and composition), and a first look at inverse and exponential functions. These ideas are the foundation for everything that follows in the chapter.

Key Concepts

Function

• A function is a rule that pairs each input with exactly one output. For $f(x) = 2x$, each $x$ maps to a single value: $f(1)=2$, $f(2)=4$, $f(10)=20$.
• Written as $y = 2x$, every $y$ is paired with one $x$. The defining property: one input → one output. If a single input can produce two outputs, it is not a function.

Evaluating a function (substitution)

• To compute $f(\text{anything})$, replace every $x$ with that input. $f(a-b+c) = 2(a-b+c)$.
• Always wrap the substituted value in parentheses to keep signs correct.

Testing whether something is a function

• Vertical line test (graphs): if every vertical line meets the graph at most once, it is a function. A curve where one $x$-value gives two heights fails.
• Algebraic test (equations): solve for $y$. If each $x$ yields exactly one $y$, it is a function; if solving produces a $\pm$ (two $y$-values), it is not.
  • $3x^2 - y = 5 \Rightarrow y = 3x^2 - 5$: one output per input — a function.
  • $y^2 - x = 5 \Rightarrow y = \pm\sqrt{x+5}$: two outputs per input — not a function.

Domain of a function

• The domain is the set of all inputs you may legally plug in. Working over the real numbers, only two things break a function:
  • Division by zero — undefined.
  • Square root of a negative number — not a real number.
• For $f(x) = \dfrac{3x-1}{x^2+x-6}$: factor the denominator, $x^2+x-6=(x+3)(x-2)$, so the domain is all reals except $x=-3$ and $x=2$.
• For $f(x) = 3x^2 - x$: no division, no square root, so the domain is all real numbers, $\mathbb{R}$.
• For $f(x) = \sqrt{2x-3}$: require $2x-3 \ge 0 \Rightarrow x \ge \tfrac{3}{2}$, so the domain is $x \ge \tfrac{3}{2}$.

Function operations

• Given functions $f$ and $g$:
  • Sum: $(f+g)(x) = f(x) + g(x)$
  • Difference: $(f-g)(x) = f(x) - g(x)$
  • Product: $(f\cdot g)(x) = f(x)\,g(x)$
  • Quotient: $\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}$, where $g(x)\neq 0$
  • Composition: $(f\circ g)(x) = f\big(g(x)\big)$ — feed $g(x)$ into $f$ as its input.
• To evaluate at a number you can either combine the functions first then substitute, or evaluate each function at the number then combine — both give the same result.

Inverse functions

• An inverse function $f^{-1}$ undoes $f$: running an input through $f$ and then $f^{-1}$ (in either order) returns the original input, $f^{-1}\big(f(x)\big) = x$ and $f\big(f^{-1}(x)\big)=x$.
• The notation $f^{-1}$ is a symbol, not an exponent ($f^{-1}\neq 1/f$).
• To find an inverse: write $y=f(x)$, swap $x$ and $y$, solve for $y$, then rename it $f^{-1}(x)$. Example: $f(x)=2x+3 \Rightarrow x=2y+3 \Rightarrow y=\dfrac{x-3}{2}$, so $f^{-1}(x)=\dfrac{x-3}{2}$.
• To verify two functions are inverses: compose them and confirm the result simplifies to exactly $x$.

Exponential functions

• An exponential function has the variable in the exponent, e.g. $f(x)=3^x$ — not $f(x)=x^3$, where the variable is the base.
• Solving by common base: rewrite both sides as powers of the same base, then set the exponents equal.
  • Useful tools: $a^m \cdot a^n = a^{m+n}$, $\left(a^m\right)^n = a^{mn}$, and $\dfrac{1}{a^n} = a^{-n}$.

Worked Example

Composition. Given $f(x) = x^2 - x$ and $g(x) = x + 3$, find $(f\circ g)(x)$.

Solving an exponential equation. Solve $8^{3x} = 32$ using a common base of $2$, since $8 = 2^3$ and $32 = 2^5$.

Summary

• A function assigns exactly one output to each input; test graphs with the vertical line test and equations by solving for $y$ and watching for a $\pm$.
• To evaluate $f$ at any expression, substitute it for every $x$ — using parentheses to protect signs.
• The domain is every real input that avoids the two breakers: division by zero and the square root of a negative number.
• Functions combine by sum, difference, product, and quotient (pointwise), and by composition $(f\circ g)(x)=f(g(x))$, which feeds one function's output into the next.
• An inverse function undoes the original: find it by swapping $x$ and $y$ and solving for $y$; verify it by composing and checking the result is just $x$.
• An exponential function keeps the variable in the exponent; solve such equations by rewriting both sides on a common base and equating the exponents.