Modules › Functions and their graphs
Functions and their graphs
Domain and range, parity, composition, transformations, the standard families of functions and modelling.
Lecture 1 · Tutorial 1 · Functions, limits and continuity
Textbook sections: Thomas ch. 1 Functions (1.1–1.3) and ch. 2 Limits and Continuity (2.1–2.6) · Stewart ch. 1 (1.1–1.3) and ch. 2 (2.1–2.6)
Textbook sections: Thomas ch. 1 Functions (1.1–1.3) and ch. 2 Limits and Continuity (2.1–2.6) · Stewart ch. 1 (1.1–1.3) and ch. 2 (2.1–2.6)
Functions and their graphs
Calculus I · Module 1
After this module you can
- Find the domain of a function given by a formula and decide whether a graph is the graph of a function.
- Decide if a function is even, odd or neither, and use the parity rules for sums, products, quotients and compositions.
- Compose functions, decompose a composite, and read a graph transformation from a formula.
- Write a quantity from a word problem as a function of one variable with its natural domain.
Function, domain and range
- A function $f:D\to\mathbb{R}$ assigns to each $x\in D$ exactly one $y=f(x)$. Vertical line test: each vertical line meets the graph at most once.
- Natural domain: all $x$ for which the formula makes sense: denominators $\ne0$, even roots $\ge0$, logarithm arguments $>0$.
- Piecewise functions: check each piece and the junction points. Example: $|y|=x$ is NOT a graph of a function (it gives $y=\pm x$).
- Range: the set of values actually taken; read it from the graph or by solving $y=f(x)$ for $x$.
Even, odd, monotone, composition
- Even: $f(-x)=f(x)$ (symmetric about the $y$-axis).
- Odd: $f(-x)=-f(x)$ (symmetric about the origin); then $f(0)=0$ if defined.
- Domain must be symmetric about $0$ first!
- $(f\circ g)(x)=f(g(x))$: apply $g$ first.
| Product / quotient | Result |
|---|---|
| even · even | even |
| odd · odd | even |
| even · odd | odd |
| $f\circ g$, $f$ even, $g$ odd | even |
| $g\circ g$, $g$ odd | odd |
Transformations of graphs
| Formula | Effect on the graph of $y=f(x)$ |
|---|---|
| $y=f(x-a)$ | shift right by $a$ (left if $a<0$) |
| $y=f(x)+b$ | shift up by $b$ |
| $y=cf(x)$ | stretch vertically by $c$ (reflect if $c<0$) |
| $y=f(cx)$ | compress horizontally by $c$ |
| $y=-f(x+4)$ | left 4, then reflect in the $x$-axis |
Inside the argument = horizontal and 'backwards'; outside = vertical.
Standard families
- Polynomials, rational functions $\frac{P}{Q}$ (undefined where $Q=0$), power and algebraic functions $x^{a}$, $\sqrt[n]{x}$.
- Trigonometric: $\sin,\cos$ (period $2\pi$), $\tan=\frac{\sin}{\cos}$, $\sec,\csc,\cot$; $\sin^2x+\cos^2x=1$.
- Exponential $a^x$ and logarithm $\log_a x$ are inverses; $\ln$ is the natural logarithm.
- Given one of $\sin,\cos,\tan$ and the quadrant, find the others with $\sin^2+\cos^2=1$ and fix the sign by the quadrant.
Modelling: a box from a sheet
- Cut squares of side $x$ from the corners of a $14\times22$ sheet and fold up the sides.
- Base $(14-2x)\times(22-2x)$, height $x$: $V(x)=x(14-2x)(22-2x)=4x^3-72x^2+308x$.
- Domain from geometry: $x>0$ and $14-2x>0$, so $0<x<7$.
- Later (Module 7) you will maximise $V$: the same function, now with calculus.
Typical mistakes
Parity without domain$f(x)=\frac{x-1}{x+1}$ is not even or odd: its domain $x\neq-1$ is not symmetric.
Order of composition$f(g(x))\ne g(f(x))$ in general: $u(v(f(x)))$ is evaluated from the inside out.
Quadrant signs$\sin x=\frac35$ in $(\frac\pi2,\pi)$ gives $\cos x=-\frac45$, not $+\frac45$.
Practice next
- Chapter quizzes Ch. 1 (3 sets)
- Practice bank, filtered to this chapter, then written mode
- Lab: Plot and derivative
- Then the mock exams (Exams page)
Worked examples from the course problems
Worked example 1 · Modelling with functions (Tutorial 1, Q6)
A sector with arc length $x$ is cut from a circular paper of radius $4$ and the rest is rolled into a cone. The base radius is $r(x)=$
Show the answer and the solution
Answer: $\dfrac{8\pi-x}{2\pi}$
The base circumference equals the remaining arc: $2\pi r=8\pi-x$, so $r=\frac{8\pi-x}{2\pi}=4-\frac{x}{2\pi}$. Slant height $4$ gives $h=\sqrt{16-r^2}$.
Worked example 2 · Even and odd functions (Computation problems 1, Q6)
Let $f$ be even and $g$ odd, both defined on all of $\mathbb{R}$. Where defined, the function $fg$ is
Show the answer and the solution
Answer: Odd
even·odd: $(fg)(-x)=f(x)\cdot(-g(x))=-(fg)(x)$.
Worked example 3 · Composition (Computation problems 1, Q4)
With $u(x)=4x-5$, $v(x)=x^2$ and $f(x)=1/x$, compute $u(v(f(x)))$.
Show the answer and the solution
Answer: $-5 + \frac{4}{x^{2}}$
Work from the inside out. $u(v(f(x)))=-5 + \frac{4}{x^{2}}$.
Worked example 4 · Transformations of graphs (Tutorial 1, Q7)
The graph of $y=f(x-4)$ is obtained from the graph of $y=f(x)$ by
Show the answer and the solution
Answer: shift $4$ units to the right
Changes inside the argument act horizontally and 'backwards' ($x-a$ moves right by $a$); changes outside act vertically.
Practice
Quiz set A Quiz set B Quiz set C Practice bank (this chapter) Written mode Lab: Plot and derivative