Question

Which statement is true regarding the graphed functions?

On a coordinate plane, a straight red line with a positive slope, labeled g of x, crosses the x-axis at (negative 6, 0) and the y-axis at (0, 6). A straight blue line with a negative slope, labeled f of x, crosses the x-axis at (negative 0.75, 0) and the y-axis at (0, negative 2).

f(4) = g(4)
f(4) = g(–2)
f(2) = g(–2)
f(–2) = g(–2)

Answer 1

To determine which statements are true regarding the functions \( f(x) \) and \( g(x) \), we need to derive the equations of both lines based on the provided information.

### Step 1: Equation of \( g(x) \)

The line \( g(x) \) has a positive slope and crosses the x-axis at (−6, 0) and the y-axis at (0, 6). We can calculate the slope using the two points:

- The coordinates of the points are:
- Point 1: \( (-6, 0) \)
- Point 2: \( (0, 6) \)

Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\[
m_g = \frac{6 - 0}{0 - (-6)} = \frac{6}{6} = 1
\]

Now, using the point-slope form \( y - y_1 = m(x - x_1) \), we can use one of the points, say (0, 6):
\[
y - 6 = 1(x - 0) \implies y = x + 6
\]

So, the equation of the function \( g(x) \) is:
\[
g(x) = x + 6
\]

### Step 2: Equation of \( f(x) \)

The line \( f(x) \) has a negative slope and crosses the x-axis at (−0.75, 0) and the y-axis at (0, −2). Again, we calculate the slope:

- The coordinates of the points are:
- Point 1: \( (-0.75, 0) \)
- Point 2: \( (0, -2) \)

Using the slope formula:
\[
m_f = \frac{-2 - 0}{0 - (-0.75)} = \frac{-2}{0.75} = -\frac{8}{3}
\]

Using the point-slope form with point (0, -2):
\[
y + 2 = -\frac{8}{3}(x - 0) \implies y = -\frac{8}{3}x - 2
\]

So, the equation of the function \( f(x) \) is:
\[
f(x) = -\frac{8}{3}x - 2
\]

### Step 3: Evaluate the statements

#### Evaluate \( f(4) \) and \( g(4) \):
\[
g(4) = 4 + 6 = 10
\]
\[
f(4) = -\frac{8}{3}(4) - 2 = -\frac{32}{3} - 2 = -\frac{32}{3} - \frac{6}{3} = -\frac{38}{3}
\]
\( f(4) \neq g(4) \)

#### Evaluate \( f(4) \) and \( g(-2) \):
\[
g(-2) = -2 + 6 = 4
\]
\( f(4) = -\frac{38}{3} \neq 4 \)

#### Evaluate \( f(2) \) and \( g(-2) \):
\[
g(-2) = -2 + 6 = 4
\]
\[
f(2) = -\frac{8}{3}(2) - 2 = -\frac{16}{3} - \frac{6}{3} = -\frac{22}{3}
\]
\( f(2) \neq g(-2) \)

#### Evaluate \( f(-2) \) and \( g(-2) \):
\[
f(-2) = -\frac{8}{3}(-2) - 2 = \frac{16}{3} - \frac{6}{3} = \frac{10}{3}
\]
\[
g(-2) = -2 + 6 = 4
\]
\( f(-2) \neq g(-2) \)

### Final Result
None of the statements provided are true:
- \( f(4) = g(4) \) is false.
- \( f(4) = g(-2) \) is false.
- \( f(2) = g(-2) \) is false.
- \( f(-2) = g(-2) \) is false.

Thus, none of the expressions listed are true regarding the values of \( f \) and \( g \).