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Read and evaluate function notation, decide a sensible domain, and interpret the intercepts, intervals, extrema and average rate of change of a graph.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you learn what $f(x)$ actually says — one output per input, with the input written inside the brackets — and how to find the set of inputs a function accepts, whether the limit comes from the rule or from the situation. Then you learn to read a graph as a story: what its intercepts, its rising and falling stretches, its highest and lowest points and its average rate of change each mean about the thing being graphed.
You already substitute a number into an expression: if $y = x^2 - 3x$ and $x = 4$, then $y = 16 - 12 = 4$. Function notation is a shorter way of saying exactly that, with a name for the rule so that several rules can be discussed at once. You also already know that you cannot take the square root of a negative number — which is where domains come from.
Function: a rule that gives exactly one output for each input.
$f(x)$: read 'f of x' — the output of the rule $f$ when the input is $x$. It is not $f$ multiplied by $x$.
Domain: the set of inputs the function accepts.
Range: the set of outputs it produces.
Argument: whatever is written inside the brackets — the input.
A function is a rule with one output per input, and $f(x)$ is the name of that output. The notation is worth having because it is precise about which number you mean: $f(3)$ is a single value, $f(x) = 7$ is a question about which inputs give $7$, and $f$ on its own is the whole rule. To evaluate, put the input in place of every $x$ — in brackets, always, so that squaring and multiplying handle its sign correctly. The domain is the set of inputs allowed. Two things limit it. The rule may forbid an input: $\sqrt{x - 4}$ needs $x \ge 4$, and $\frac{1}{x - 2}$ forbids $x = 2$. And the situation may forbid one: a function giving the cost of $n$ tickets has the whole numbers as its domain, because there is no such purchase as $2.5$ tickets.
Another way: diagram
A machine box labelled $f$ with an arrow in from a tray labelled 'domain: $x \ge 4$' and an arrow out to a tray labelled 'range', and the single input $13$ shown going in and $3$ coming out.
Another way: story
Think of $f$ as a machine and the brackets as its input slot. The domain is what the slot will accept — some things jam it, and some things (half a ticket) never turn up in the first place.
"$f(x)$ means $f$ times $x$." It does not. The brackets say "the output at", and $f(2 + 3)$ is $f(5)$, not $f(2) + f(3)$.
"$g(-2) = -2^2 - 3(-2)$, so $-4 + 6 = 2$." Put brackets round the input: $(-2)^2 = 4$, so $g(-2) = 4 + 6 = 10$. The missing brackets cost the sign.
"The domain is always all real numbers." Two things cut it down: the rule itself (no negative under a square root, no dividing by zero) and the situation (you cannot buy $2.5$ tickets).
Replace every $x$ with $(-2)$, brackets included: $(-2)^2 - 3(-2)$.
The brackets are what protect the sign.
$(-2)^2 = 4$ and $-3 \times (-2) = +6$.
A negative times a negative is positive.
So $g(-2) = 4 + 6 = 10$.
The root demands that what is inside be zero or more: $x - 4 \ge 0$.
Real square roots need a non-negative radicand.
So $x \ge 4$: the domain is every number from $4$ upwards.
What is inside the root must be at least zero: $x - 2 \ge 0$.
So the domain is $x \ge 2$.
If $g(x) = x^2 - 5x$, what is $g(-1)$?
answer
What is the domain of $f(x) = \sqrt{x - 9}$?
A function gives the cost of $n$ concert tickets at $\$8$ each: $C(n) = 8n$. What is a sensible domain?
Match each term to what it means.
| the set of allowed inputs | the set of outputs | the output when the input is $8$ | the inputs whose output is $4$ | |
|---|---|---|---|---|
| the domain | ||||
| the range | ||||
| $f(8)$ | ||||
| the solutions of $f(x) = 4$ |
A rectangle has a perimeter of $20$ m. If one side is $x$ m, the other is $10 - x$ m, so the area is $A(x) = x(10 - x)$. Which values of $x$ belong in the domain?
You already find where a line crosses an axis, and you already compute a slope as rise over run. This lesson gives names to the parts of a graph that matter and asks what each one means in the situation the graph describes — which is the difference between reading a graph and merely looking at one.
Intercept: where the graph meets an axis. The $y$-intercept is the output at input $0$; an $x$-intercept, or zero, is an input whose output is $0$.
Increasing / decreasing: the output rises / falls as you move right.
Maximum / minimum: the largest / smallest output, often at a turning point.
Average rate of change from $a$ to $b$: $\frac{f(b) - f(a)}{b - a}$ — the slope of the straight line joining the two points.
Line of symmetry: a vertical line the graph is a mirror image about.
The features of a graph each answer a question about the situation. The $y$-intercept is the starting value — what the quantity is before anything happens. A zero is an input at which the quantity is nothing at all: for a profit function that is the break-even point, and for a height it is the ground. The intervals where the graph is increasing or decreasing say when the quantity is growing or shrinking, and a maximum or minimum marks the turn between them. The average rate of change over an interval, $\frac{f(b) - f(a)}{b - a}$, is how fast the quantity changed on average, and it carries the units of output per unit of input — dollars per item, metres per second. Reading these off is only half the job; saying what each one means in the situation is the other half.
Another way: diagram
A curve of height against time with the $y$-intercept labelled 'launch height', the peak labelled 'maximum', the crossing of the time axis labelled 'lands here', and a dashed chord between two points labelled 'average rate of change'.
Another way: story
Every graph is a sentence about something. Before naming the feature, say aloud what happens there — 'this is where there is none left', 'this is the highest it ever gets' — and the technical word follows.
"Increasing means the values are positive." No: it means they are getting larger. A temperature going from $-10$ to $-3$ is increasing the whole way.
"An intercept of $20$ means the maximum is $20$." An $x$-intercept is an input. If profit crosses zero at $n = 20$, then $20$ is a number of items, not an amount of money.
"Average rate of change is the average of the two values." It is a slope: the change in output divided by the change in input. $\frac{30 - 10}{6 - 2} = 5$, not $20$.
The output changed by $30 - 10 = 20$.
Change in output on top.
The input changed by $6 - 2 = 4$.
Change in input underneath.
So the average rate of change is $\frac{20}{4} = 5$ output units per input unit.
On that axis the output is zero, so profit is zero when $20$ items are sold.
An intercept is a place where a value is zero.
Zero profit means costs and takings match: it is the break-even point.
Translate the feature back into the situation.
On the $x$-axis, $y = 0$, so $3x - 9 = 0$.
So $3x = 9$ and $x = 3$: the graph crosses at $(3, 0)$.
For a function $f$, $f(6) = 14$ and $f(11) = 24$. What is the average rate of change from $6$ to $11$?
answer
A company's profit $P(n)$ for selling $n$ items crosses the $n$-axis at $n = 47$. What does that mean?
A ball's height against time rises until $t = 2$ seconds and falls after that. On which interval is the height increasing?
A function $f$ has the four properties below. Match each to the feature it describes.
| $0$ is the $y$-intercept | $28$ is the maximum | $8$ is a zero | $x = 2$ is the line of symmetry | |
|---|---|---|---|---|
| $f(0) = 0$ | ||||
| $f(2) = 28$, and $f(x) \le 28$ for every $x$ | ||||
| $f(8) = 0$ | ||||
| $f(2 - t) = f(2 + t)$ for every $t$ |
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
If $f(x) = 8x + 4$, what is $f(-1)$?
answer
Where does $y = 9x - 36$ cross the $x$-axis? Give the $x$ value.
Answer:
You can evaluate a function, state a sensible domain, and say what a graph's key features mean. Explain why $g(-2) = 10$ when $g(x) = x^2 - 3x$, and what it means for a profit graph to cross the axis at $20$ items.
8. Your turn: the domain of $f(x) = \sqrt{x - 2}$, step 2
20. Your turn: where does $y = 3x - 9$ cross the $x$-axis?, step 2