Linear Function Problem Overview

A linear function is a function whose graph is a straight line. It can be represented by the equation \(f(x) = mx + c\), where \(m\) is the slope of the line and \(c\) is the y-intercept. In this problem, we are given two specific values of the function at two different x-values and asked to find the value of the function at another x-value.

Calculating the Slope of the Linear Function

The first step to finding the equation of the linear function is to determine its slope (\(m\)). The slope is calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

We are given two points on the line:

• Point 1: \(x_1 = -2\), \(f(x_1) = y_1 = 29\)
• Point 2: \(x_2 = 3\), \(f(x_2) = y_2 = 39\)

Now, let's substitute these values into the slope formula:

$$m = \frac{39 - 29}{3 - (-2)}$$ $$m = \frac{10}{3 + 2}$$ $$m = \frac{10}{5}$$ $$m = 2$$

So, the slope of the linear function is \(2\).

Determining the Equation of the Linear Function

Now that we have the slope (\(m = 2\)), we can use one of the given points to find the y-intercept (\(c\)) of the linear function. We will use the general form \(f(x) = mx + c\). Let's use the point \((x, f(x)) = (-2, 29)\):

$$f(x) = mx + c$$ $$29 = (2)(-2) + c$$ $$29 = -4 + c$$

To find \(c\), we add \(4\) to both sides of the equation:

$$c = 29 + 4$$ $$c = 33$$

Thus, the equation of the linear function is:

$$f(x) = 2x + 33$$

Finding the Function Value at x = 5

The final step is to find the value of the linear function at \(x = 5\). We will substitute \(x = 5\) into the equation \(f(x) = 2x + 33\):

$$f(5) = 2(5) + 33$$ $$f(5) = 10 + 33$$ $$f(5) = 43$$

Therefore, the value of the linear function at \(x = 5\) is \(43\).

Summary of Steps for Linear Function Value

Here is a quick summary of how we found the value of the linear function:

The final value of the linear function at \(x = 5\) is \(43\).