Matemática Elementar

Problema 1

Let $f(x) = 3x - 7$ . Evaluate each of the following.

a) $f(4)$

b) $f(-2)$

c) $f(0)$

Problemas

1. Let $f(x) = 3x - 7$ . Evaluate each of the following.

a) $f(4)$

b) $f(-2)$

c) $f(0)$

2. A function $g$ is defined by $g(x) = -2x + 10$ . Find the value of $x$ such that $g(x) = 0$ .

3. The table below shows values of a linear function $h(x)$ .

| $x$ | $-1$ | $0$ | $1$ | $2$ | $3$ | |--------|------|-----|-----|-----|-----| | $h(x)$ | $8$ | $5$ | $2$ | $-1$ | $-4$ |

a) What is the $y$ -intercept of $h$ ?

b) What is the rate of change (slope) of $h$ ?

c) Write the equation for $h(x)$ .

4. The graph of a linear function $p(x)$ passes through the points $(0, -3)$ and $(6, 9)$ .

a) Find the slope of $p$ .

b) State the domain and range of $p$ if it is defined for $-2 \leq x \leq 8$ .

c) Find $p(5)$ .

5. A car rental company charges a flat fee of $\$ 30 $plus$ $0.20 $per mile driven. The total cost in dollars is modeled by$ C(m) = 0.20m + 30 $, where$ m$ is the number of miles driven.

a) Evaluate $C(100)$ and explain what it means in context.

b) Find the value of $m$ such that $C(m) = 70$ . What does this represent?

c) What is the $y$ -intercept of $C$ , and what does it mean in the context of this problem?

Gabarito — Linear Functions

1. a) $f(4) = 3(4) - 7 = 12 - 7 = 5$

b) $f(-2) = 3(-2) - 7 = -6 - 7 = -13$

c) $f(0) = 3(0) - 7 = 0 - 7 = -7$

2. Set $g(x) = 0$ and solve:

$-2x + 10 = 0$ $-2x = -10$ $x = 5$

3. a) The $y$ -intercept is the value of $h(x)$ when $x = 0$ . From the table, $h(0) = 5$ , so the $y$ -intercept is $5$ .

b) The rate of change (slope) is constant for a linear function. Using any two consecutive points:

$m = \frac{h(1) - h(0)}{1 - 0} = \frac{2 - 5}{1} = -3$

The slope is $-3$ .

c) Using slope-intercept form with $m = -3$ and $b = 5$ :

$h(x) = -3x + 5$

4. a) The slope is:

$m = \frac{9 - (-3)}{6 - 0} = \frac{12}{6} = 2$

b) Since $p$ is linear and defined for $-2 \leq x \leq 8$ :

Domain: $[-2, 8]$

To find the range, evaluate $p$ at the endpoints. With slope $2$ and $y$ -intercept $-3$ , we have $p(x) = 2x - 3$ .

$p(-2) = 2(-2) - 3 = -7$ $p(8) = 2(8) - 3 = 13$

Range: $[-7, 13]$

c) $p(5) = 2(5) - 3 = 10 - 3 = 7$

5. a) $C(100) = 0.20(100) + 30 = 20 + 30 = 50$

This means driving 100 miles costs $\$ 50$ in total.

b) $0.20m + 30 = 70$ $0.20m = 40$ $m = 200$

This means you can drive 200 miles for a total cost of $\$ 70$.

c) The $y$ -intercept is $C(0) = 30$ . It represents the flat fee charged before any miles are driven — the base cost of the rental.