Relations and Functions: Practice

The root sits in a denominator, so its inside must be defined and nonzero, meaning strictly positive.

$4 - x > 0 \;\Rightarrow\; x < 4$

At $x = 4$ the denominator would be $\sqrt{0} = 0$, which is not allowed, so the domain is $x < 4$.

On this domain the square reaches its smallest value at $x = 0$ and its largest at $x = 3$.

$0^2 = 0 \quad \text{and} \quad 3^2 = 9$

Since $x = 0$ lies in the domain, the minimum output is $0$, not $1$. The range is $0 \le y \le 9$.

Every output lands in the declared codomain, but not every element of the codomain must be used.

$\text{range} \subseteq \text{codomain}$

So the range always sits inside the codomain and may be smaller.

One-to-one means no two inputs share an output, so avoid pairing $c$ with $-c$.

$a, b \ge 0, \; a^2 = b^2 \;\Rightarrow\; a = b$

On $x \ge 0$ the square is strictly increasing and one-to-one. Any interval containing both $c$ and $-c$ fails, since they share the output $c^2$.

Both roots must have nonnegative insides at the same time.

$x - 2 \ge 0 \text{ and } 5 - x \ge 0 \;\Rightarrow\; x \ge 2 \text{ and } x \le 5$

The overlap is $2 \le x \le 5$, the domain of the sum.

Swapping gives $\{(2,1),(4,2),(6,3)\}$; test whether any first coordinate repeats.

$\{(2,1),(4,2),(6,3)\} \text{ has distinct first coordinates}$

Because the original outputs $2, 4, 6$ were all different, the swapped set has no repeated input and is a function.

The root needs a nonnegative inside, and the denominator cannot be zero.

$x + 3 \ge 0 \text{ and } x \ne 0 \;\Rightarrow\; x \ge -3 \text{ and } x \ne 0$

Note that $x = 0$ satisfies the root condition but kills the denominator, so it is still excluded.

Test both readings of the circle.

$x = 0 \Rightarrow y = \pm 3, \qquad y = 0 \Rightarrow x = \pm 3$

Both a vertical and a horizontal line can cut it twice, so it is a function of neither variable.

Apply the rule to each input, then collect the distinct outputs.

$\{(-2)^2,(-1)^2,0^2,1^2,2^2\} = \{4,1,0,1,4\} = \{0,1,4\}$

The range is the set of outputs actually produced, $\{0,1,4\}$.

The vertical line test checks whether reading $x$ as the input gives a function. Swapping which variable is the input turns vertical lines into horizontal ones.

$\text{vertical line test for } y \text{ as input} = \text{horizontal line test}$

So the horizontal line test is the same test applied after swapping $x$ and $y$.

The range is the set of outputs actually produced, which for a square is the nonnegative numbers.

$x^2 = -4 \text{ has no real solution}$

So $-4$ lives in the real-number codomain but is never an output; it is not in the range.

Two functions differ when their rule or their domain differs. Compare the domains.

$\text{domain } [0, \infty) \;\ne\; \text{all reals}$

The pair with the same rule $2x$ but different domains are different functions. The other pairs share both rule and domain, so each pair is a single function written two ways.