Domain and Range Practice

These questions follow Domain and Range. For each function, give both sets and — more importantly — the reason each restriction exists. A stated reason is worth more than a memorised answer.

From equations

1. $f(x)=x^2-3$
2. $f(x)=\sqrt{x-4}$
3. $f(x)=\frac{1}{x+2}$
4. $f(x)=2\sqrt{x}+1$

Answer 1

Domain: all real numbers — squaring accepts anything. Range: $y\ge -3$, because $x^2\ge 0$ always, and the whole graph is the parent parabola slid down 3.

Answer 2

Domain: $x\ge 4$ — the root refuses negative input, so $x-4\ge 0$. Range: $y\ge 0$, since a square root never returns a negative. The parent’s sets slid right 4 along with the graph.

Answer 3

Domain: $x\ne -2$ — division by zero at exactly one input. Range: $y\ne 0$ — a fraction with numerator 1 can shrink toward zero forever but never arrive. Both restrictions are the parent $\frac{1}{x}$‘s, with the vertical one slid left 2.

Answer 4

Domain: $x\ge 0$. Range: $y\ge 1$ — the parent’s outputs ($y\ge 0$) are stretched by 2 (still $\ge 0$) and then lifted by 1.

From contexts

5. A ball’s height is $h(t)=-5t^2+20t$ metres after $t$ seconds. State the domain and range of the model, not of the algebra.
6. Movie tickets cost $14 each, so a group’s cost is $C(n)=14n$. What are the domain and range, and what makes this function different in kind from the ones above?

Answer 5

The zeros of $h$ are $t=0$ and $t=4$ (factor: $-5t(t-4)$), so the flight lasts from $t=0$ to $t=4$: domain $0\le t\le 4$. The peak is midway, at $t=2$, where $h(2)=20$: range $0\le h\le 20$. Algebra offered all real numbers; the physics declined.

Answer 6

Domain: $n\in \{0,1,2,3,\dots \}$ — you cannot buy 2.7 tickets. Range: $\{0,14,28,42,\dots \}$. This is a discrete function: its graph is separated dots, a kind you will meet again in Sequences and Their Rules.

Stretch

7. State the domain and range of $g(x)=-\sqrt{x}$, and explain how they differ from those of $\sqrt{x}$.
8. State the domain and range of $f(x)=\frac{3}{x-1}+2$.

Answer 7

Domain: $x\ge 0$ — the reflection is vertical, so allowed inputs do not change. Range: $y\le 0$ — every output of $\sqrt{x}$ got its sign flipped. Reflections in the $x$-axis rewrite the range and leave the domain alone.

Answer 8

The parent $\frac{1}{x}$ slid right 1 and up 2 (the 3 stretches but forbids nothing new). Domain: $x\ne 1$; range: $y\ne 2$. The excluded values are exactly the new asymptotes — sketch it or confirm in Using Desmos, and the two gaps stare back at you.