Chapter at a Glance
- Chapter: 1
- Strand: Pure — Change
- Difficulty: Level 4 of 5
- Builds on: Inverse functions and composition, Graphing functions
Try This First
Try this short activity before reading the lesson.
> Take $f(x) = x^2$ and $g(x) = x - 3$. On paper, work out $f(g(5))$ and $g(f(5))$. They are not the same number. Then try to find one input where they are equal.
Math Lab: Function machines — order matters
Predict first. Let $f(x)=x^2$ and $g(x)=x+1$. Will squaring then adding 1 match adding 1 then squaring?
<div class="math-model"><p><strong>Machine A: g(f(2)) = 5</strong></p><div class="model-bar"><span class="model-cell">Input 2</span><span class="model-cell">Square: 4</span><span class="model-cell">Add 1: 5</span></div><p class="model-caption">Machine B: 2 → add 1 → 3 → square → 9.</p></div>
Model and solve. $g(f(x))=x^2+1$, while $f(g(x))=(x+1)^2$. At $x=2$, the outputs are 5 and 9. The order changes the function.
Spot the bug / explain. Why does $(x+1)^2=x^2+1$ fail? Expand and identify the missing term.
Your turn. At which real input do these two compositions have the same output?
Check after trying. $x^2+1=x^2+2x+1$ gives $x=0$.
The Big Idea
A function is a rule, and this topic is about reading the rule itself rather than one output at a time. Where is the rule allowed to run (domain)? What can it produce (range)? Does it climb or fall on a stretch of inputs? Is it symmetric?
Two functions can be chained: $f \circ g$ means "do $g$ first, then $f$". The chain runs only where $g$ runs and $g$'s output is legal for $f$. That second check is the whole difficulty of composition. An inverse is the rule run backwards, and it exists only when the forward rule never sends two inputs to one output.
Everything here comes back in calculus. The average rate of change is a slope over an interval, and the derivative will be that slope over a shrinking interval.
In brief. A function read before it is computed with: domain, range, even and odd symmetry, where it rises and falls, and its average rate of change. Piecewise rules, composition $f\circ g$ with its domain found from $g$ first, and inverses verified by composing.
Words to Know
- domain: Every input the rule accepts. Square roots need a non-negative inside; fractions need a non-zero bottom.
- even, odd: Even: $f(-x) = f(x)$, mirror in the $y$-axis. Odd: $f(-x) = -f(x)$, half-turn about the origin. Most functions are neither.
- composition: $(f \circ g)(x) = f(g(x))$. The inner function runs first.
- average rate of change: $\frac{f(b) - f(a)}{b - a}$: the slope of the line through two points of the graph.
Formulas and Theorems
- Average rate of change: $\frac{f(b)-f(a)}{b-a}$
- Note: The slope of the secant line through $(a,f(a))$ and $(b,f(b))$.
- Composition: $(f\circ g)(x)=f(g(x))$
- Note: Its domain is every $x$ in the domain of $g$ whose output $g(x)$ is in the domain of $f$.
- Inverse check: $f(f^{-1}(x))=x \text{ and } f^{-1}(f(x))=x$
- Note: Both compositions must give $x$ back.
Outcomes and Milestones
- Students can: state domain and range from a rule or a graph; test even/odd algebraically; find intervals of increase and decrease; compute average rate of change; evaluate and graph piecewise functions; build $f\circ g$ and find its domain; find $f^{-1}$ and verify by composition.
- Watch for: the domain of $f\circ g$ read from the simplified formula instead of from $g$'s domain first; “odd” checked by plugging in one number; average rate of change with the inputs swapped in only one of numerator and denominator.
- Ready when: for $f(x)=\sqrt{x-1}$ and $g(x)=\frac{1}{x}$, the domain of $f\circ g$ is $(0,1]$ with a reason, and $f^{-1}$ is written with its domain.
Worked Examples and Your Turn
Study each example, then try the paired problem.
Example 1: Domain from the rule alone
Problem: Find the domain of $h(x) = \dfrac{\sqrt{5 - x}}{x + 3}$.
Solution:
- Inside the root: $5 - x \ge 0$, so $x \le 5$. Why: a square root of a negative number is not a real number.
- Bottom of the fraction: $x + 3 \ne 0$, so $x \ne -3$. Why: division by zero is undefined.
- Combine both: $x \le 5$ and $x \ne -3$. Why: an input must pass every test at once.
Answer: $(-\infty, -3) \cup (-3, 5]$
> Verification & Check: Test $x = 0$: $\frac{\sqrt5}{3}$, fine. Test $x = 6$: root of $-1$, excluded. Test $x = -3$: bottom is zero, excluded.
Your Turn:
- Prompt: Domain of $k(x) = \dfrac{x}{\sqrt{x - 2}}$.
- Answer: $x > 2$, written $(2, \infty)$. The root is on the bottom, so $x - 2 = 0$ is out as well.
Example 2: Odd or even, by algebra
Problem: Decide whether $f(x) = \dfrac{x^3}{x^2 + 1}$ is even, odd, or neither.
Solution:
- Replace $x$ by $-x$: $f(-x) = \dfrac{(-x)^3}{(-x)^2 + 1} = \dfrac{-x^3}{x^2 + 1}$. Why: the definition is about $f(-x)$ for every $x$, so substitute symbolically.
- Compare with $f(x)$: $f(-x) = -f(x)$. Why: the top flipped sign and the bottom did not, so the whole thing flipped.
Answer: Odd.
> Notice: Plugging in one number, say $x = 2$, can show a function is not even or odd. It can never show that it is. Only the algebra covers every $x$.
Your Turn:
- Prompt: Even, odd, or neither: $g(x) = x^4 - 2x^2 + 7$?
- Answer: Even. Every power of $x$ is even, so $g(-x) = g(x)$.
Example 3: Average rate of change on an interval
Problem: Let $f(x) = 2^x$. Find the average rate of change from $x = 1$ to $x = 4$.
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 380 270" width="380" height="270" role="img" aria-label="Graph of y = 2 to the x from about x = −0.6 to 4.2. Points (1, 2) and (4, 16) are marked on the curve and joined by a red secant line.
The chapter continues in the book.
