3.5 Relations and Functions

Before you get started, take this readiness quiz.

Evaluate $3 x - 5$ when $x = −2$.
If you missed this problem, review the reference.

Evaluate $2 x^{2} - x - 3$ when $x = a .$
If you missed this problem, review the reference.

Simplify: $7 x - 1 - 4 x + 5.$
If you missed this problem, review the reference.

For the relation $\left\{\left(1 , 1\right) , \left(2 , 4\right) , \left(3 , 9\right) , \left(4 , 16\right) , \left(5 , 25\right)\right\} :$

ⓐ Find the domain of the relation.

ⓑ Find the range of the relation.

For the relation $\left\{\left(1 , 1\right) , \left(2 , 8\right) , \left(3 , 27\right) , \left(4 , 64\right) , \left(5 , 125\right)\right\} :$

ⓐ Find the domain of the relation.

ⓑ Find the range of the relation.

For the relation $\left\{\left(1 , 3\right) , \left(2 , 6\right) , \left(3 , 9\right) , \left(4 , 12\right) , \left(5 , 15\right)\right\} :$

ⓐ Find the domain of the relation.

ⓑ Find the range of the relation.

Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the relation.

ⓐ $\left\{\left(−3 , 27\right) , \left(−2 , 8\right) , \left(−1 , 1\right) , \left(0 , 0\right) , \left(1 , 1\right) , \left(2 , 8\right) , \left(3 , 27\right)\right\}$

ⓑ $\left\{\left(9 , −3\right) , \left(4 , −2\right) , \left(1 , −1\right) , \left(0 , 0\right) , \left(1 , 1\right) , \left(4 , 2\right) , \left(9 , 3\right)\right\}$

Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the function.

ⓐ $\left\{\left(−3 , −6\right) , \left(−2 , −4\right) , \left(−1 , −2\right) , \left(0 , 0\right) , \left(1 , 2\right) , \left(2 , 4\right) , \left(3 , 6\right)\right\}$

ⓑ $\left\{\left(8 , −4\right) , \left(4 , −2\right) , \left(2 , −1\right) , \left(0 , 0\right) , \left(2 , 1\right) , \left(4 , 2\right) , \left(8 , 4\right)\right\}$

Use the set of ordered pairs to (i) determine whether the relation is a function (ii) find the domain of the relation (iii) find the range of the relation.

ⓐ $\left\{\left(27 , −3\right) , \left(8 , −2\right) , \left(1 , −1\right) , \left(0 , 0\right) , \left(1 , 1\right) , \left(8 , 2\right) , \left(27 , 3\right)\right\}$

ⓑ $\left\{\left(7 , −3\right) , \left(−5 , −4\right) , \left(8 , 0\right) , \left(0 , 0\right) , \left(−6 , 4\right) , \left(−2 , 2\right) , \left(−1 , 3\right)\right\}$

Determine whether each equation is a function.

ⓐ $4 x + y = −3$ ⓑ $x + y^{2} = 1$ ⓒ $y - x^{2} = 2$

Determine whether each equation is a function.

ⓐ $x + y^{2} = 4$ ⓑ $y = x^{2} - 7$ ⓒ $y = 5 x - 4$

For the function $f \left(x\right) = 3 x^{2} - 2 x + 1 ,$ evaluate the function.

ⓐ $f \left(3\right)$ ⓑ $f \left(−1\right)$ ⓒ $f \left(t\right)$

For the function $f \left(x\right) = 2 x^{2} + 4 x - 3 ,$ evaluate the function.

ⓐ $f \left(2\right)$ ⓑ $f \left(−3\right)$ ⓒ $f \left(h\right)$

For the function $g \left(x\right) = 4 x - 7 ,$ evaluate the function.

ⓐ $g \left(m^{2}\right)$ ⓑ $g \left(x - 3\right)$ ⓒ $g \left(x\right) - g \left(3\right)$

For the function $h \left(x\right) = 2 x + 1 ,$ evaluate the function.

ⓐ $h \left(k^{2}\right)$ ⓑ $h \left(x + 1\right)$ ⓒ $h \left(x\right) + h \left(1\right)$

The number of unread emails in Bryan’s account is 100. This number grows by 15 unread emails a day. The function $N \left(t\right) = 100 + 15 t$ represents the relation between the number of emails, N, and the time, t, measured in days.

ⓐ Determine the independent and dependent variable.

ⓑ Find $N \left(7\right) .$ Explain what this result means.

The number of unread emails in Anthony’s account is 110. This number grows by 25 unread emails a day. The function $N \left(t\right) = 110 + 25 t$ represents the relation between the number of emails, N, and the time, t, measured in days.

ⓐ Determine the independent and dependent variable.

ⓑ Find $N \left(14\right) .$ Explain what this result means.

For each relation ⓐ find the domain of the relation ⓑ find the range of the relation.

$\left\{\right. \left(1 , 4\right) , \left(2 , 8\right) , \left(3 , 12\right) , \left(4 , 16\right) , \left(5 , 20\right) \left.\right\}$

For each relation ⓐ find the domain of the relation ⓑ find the range of the relation.

$\left\{\left(1 , 7\right) , \left(5 , 3\right) , \left(7 , 9\right) , \left(−2 , −3\right) , \left(−2 , 8\right)\right\}$

Use the set of ordered pairs to ⓐ determine whether the relation is a function, ⓑ find the domain of the relation, and ⓒ find the range of the relation.

$\left\{\right. \left(−3 , 9\right) , \left(−2 , 4\right) , \left(−1 , 1\right) ,$
$\left(0 , 0\right) , \left(1 , 1\right) , \left(2 , 4\right) , \left(3 , 9\right) \left.\right\}$

Use the set of ordered pairs to ⓐ determine whether the relation is a function, ⓑ find the domain of the relation, and ⓒ find the range of the relation.

$\left\{\right. \left(−3 , 27\right) , \left(−2 , 8\right) , \left(−1 , 1\right) ,$
$\left(0 , 0\right) , \left(1 , 1\right) , \left(2 , 8\right) , \left(3 , 27\right) \left.\right\}$

Determine whether each equation is a function.

ⓐ $2 x + y = −3$
ⓑ $y = x^{2}$
ⓒ $x + y^{2} = −5$

Determine whether each equation is a function.

ⓐ $y - 3 x^{3} = 2$
ⓑ $x + y^{2} = 3$
ⓒ $3 x - 2 y = 6$

Evaluate the function: ⓐ $f \left(2\right)$ ⓑ $f \left(−1\right)$ ⓒ $f \left(a\right) .$

$f \left(x\right) = 5 x - 3$

Evaluate the function: ⓐ $f \left(2\right)$ ⓑ $f \left(−1\right)$ ⓒ $f \left(a\right) .$

$f \left(x\right) = −4 x + 2$

Evaluate the function: ⓐ $f \left(2\right)$ ⓑ $f \left(−1\right)$ ⓒ $f \left(a\right) .$

$f \left(x\right) = x^{2} - x + 3$

Evaluate the function: ⓐ $f \left(2\right)$ ⓑ $f \left(−1\right)$ ⓒ $f \left(a\right) .$

$f \left(x\right) = 2 x^{2} - x + 3$

Evaluate the function: ⓐ $g \left(h^{2}\right)$ ⓑ $g \left(x + 2\right)$ ⓒ $g \left(x\right) + g \left(2\right) .$

$g \left(x\right) = 2 x + 1$

Evaluate the function: ⓐ $g \left(h^{2}\right)$ ⓑ $g \left(x + 2\right)$ ⓒ $g \left(x\right) + g \left(2\right) .$

$g \left(x\right) = −3 x - 2$

Evaluate the function: ⓐ $g \left(h^{2}\right)$ ⓑ $g \left(x + 2\right)$ ⓒ $g \left(x\right) + g \left(2\right) .$

$g \left(x\right) = 3 - x$

Evaluate the function.

$f \left(x\right) = 3 x^{2} - 5 x ;$ $f \left(2\right)$

Evaluate the function.

$F \left(x\right) = 2 x^{2} - 3 x + 1 ;$
$F \left(−1\right)$

Evaluate the function.

$h \left(t\right) = 2 \left|t - 5\right| + 4 ;$ $h \left(−4\right)$

Evaluate the function.

$f \left(x\right) = \frac{x + 2}{x - 1} ;$ $f \left(2\right)$

Solve.

The number of unwatched shows in Sylvia’s DVR is 85. This number grows by 20 unwatched shows per week. The function $N \left(t\right) = 85 + 20 t$ represents the relation between the number of unwatched shows, N, and the time, t, measured in weeks.

ⓐ Determine the independent and dependent variable.

ⓑ Find $N \left(4\right) .$ Explain what this result means

Solve.

The daily cost to the printing company to print a book is modeled by the function $C \left(x\right) = 3.25 x + 1500$ where C is the total daily cost in dollars and x is the number of books printed.

ⓐ Determine the independent and dependent variable.

ⓑ Find $C \left(0\right) .$ Explain what this result means.

ⓒ Find $C \left(1000\right) .$ Explain what this result means.