Function Notation Practice

These questions follow What Is a Function and Function Notation — first the judgement call, then the language. Exact answers throughout; no calculator needed.

Function or not

1. Decide whether each relation is a function, and say how you know: (a) $\{(1,3),(2,5),(3,7)\}$; (b) $\{(4,2),(4,-2),(9,3)\}$.
2. Is $x=y^2$ a function? Support your answer with actual numbers, not just a rule quoted from memory.

Answer 1

(a) A function — every input appears once, so no input has two outputs. (b) Not a function — the input 4 appears with outputs 2 and $-2$. One splitting input is all it takes; the well-behaved $(9,3)$ cannot rescue it.

Answer 2

Not a function. Feed in $x=4$: both $y=2$ and $y=-2$ satisfy $4=y^2$, so one input produces two outputs. Graphically, the sideways parabola fails the vertical-line test at every positive $x$.

Evaluate

3. For $g(x)=3x-5$, find $g(4)$, $g(0)$, and $g\left(\frac{2}{3}\right)$.
4. For $f(x)=2x^2+3x-1$, find $f\left(\frac{1}{2}\right)$.
5. Same $f$: find $f(-2)$, then explain why getting the same value as question 4 breaks no rules.

Answer 3

$g(4)=12-5=7$; $g(0)=-5$; $g\left(\frac{2}{3}\right)=2-5=-3$. Each input goes into the rule wrapped in brackets, and everything else follows.

Answer 4

$f\left(\frac{1}{2}\right)=2\left(\frac{1}{4}\right)+\frac{3}{2}-1=\frac{1}{2}+\frac{3}{2}-1=1$. The most common slip is squaring $\frac{1}{2}$ to $\frac{1}{2}$ — brackets first, then arithmetic.

Answer 5

$f(-2)=2(4)-6-1=1$. Two inputs sharing an output is many-to-one, which functions allow; only one input with two outputs is forbidden. (Both points, $\left(\frac{1}{2},1\right)$ and $(-2,1)$, sit at the same height on the parabola — one on each side of the vertex.)

Interpret and work backwards

6. The height of a ball is modelled by $h(t)=-5t^2+20t+1$, with $h$ in metres and $t$ in seconds. Compute $h(2)$, then write what the statement $h(2)=\underline{}$ means as a plain sentence.
7. For $g(x)=3x-5$, solve $g(x)=16$.
8. For $f(x)=x^2-4$, find all inputs with $f(x)=21$.

Answer 6

$h(2)=-20+40+1=21$. Sentence: two seconds after the throw, the ball is 21 metres above the ground. The notation packs the input, the output, and their meanings into one statement.

Answer 7

$3x-5=16$ gives $3x=21$, so $x=7$. Note the reversal: question 3 turned inputs into outputs; this one starts from the output — an equation to solve, not a substitution to perform.

Answer 8

$x^2-4=21$ gives $x^2=25$, so $x=5$ or $x=-5$. The question said all inputs for a reason — a quadratic can reach the same output from two sides, and reporting only one is the most common miss here.