Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Work each problem related to linear functions. (a) Evaluate \(f(-2)\) and \(f(4)\) (b) Graph \(f\). How can the graph of \(f\) be used to determine the zero of \(f ?\) (c) Find the zero of \(f\) $$f(x)=x+2$$

Step-by-Step Solution

Answer

(a) \( f(-2) = 0 \), \( f(4) = 6 \). (b) The graph intersects the x-axis at \( x = -2 \). (c) The zero of \( f \) is \( x = -2 \).

Step 1: Evaluate f(-2)

To evaluate \( f(-2) \), substitute \( -2 \) into the function \( f(x) = x + 2 \). This gives: \[ f(-2) = (-2) + 2 = 0 \] Therefore, \( f(-2) = 0 \).

Step 2: Evaluate f(4)

To evaluate \( f(4) \), substitute \( 4 \) into the function \( f(x) = x + 2 \). This gives: \[ f(4) = 4 + 2 = 6 \] Therefore, \( f(4) = 6 \).

Step 3: Graph the function f(x)

To graph \( f(x) = x + 2 \), plot two points and draw a line through them. Use the points found in previous steps: \((-2, 0)\) and \((4, 6)\). Connect these two points with a straight line. This line represents the graph of the function \( f(x) \).

Step 4: Understand graph to find zero of f

The zero of a function is where its graph intersects the x-axis. From the graph, we see that the line crosses the x-axis at \( x = -2 \), confirming our calculation from Step 1. This intersection point is the zero of the function \( f(x) = x + 2 \).

Step 5: Solve algebraically for zero of f

To find the zero, set \( f(x) = 0 \): \[ x + 2 = 0 \] Subtract 2 from both sides, \[ x = -2 \] Thus, the zero of \( f \) is \( x = -2 \).

Key Concepts

Evaluating Functions

Evaluating a function means substituting a specific value of the variable into the function to determine the corresponding output. Let's take the function \( f(x) = x + 2 \) as our example. To evaluate \( f(-2) \), simply replace \( x \) with \( -2 \) in the function. This leads to the calculation:

• \( f(-2) = (-2) + 2 = 0 \)

This tells us that when \( x \) is \(-2\), the output is \(0\). Similarly, to evaluate \( f(4) \), we substitute:

• \( f(4) = 4 + 2 = 6 \)

This means when \( x \) is \(4\), the output is \(6\). By evaluating functions at specific points, you gain insights into how the function behaves for these values.

Graphing Functions

Graphing a function involves plotting points on a coordinate plane and connecting them to show the relationship between the input \( x \) and the output \( f(x) \). For the linear function \( f(x) = x + 2 \), you can start by plotting a few calculated points. We already know from our evaluations:

• \(-2, 0\)
• \(4, 6\)

Plot these points on a graph. Draw a straight line connecting them, extending the line beyond both points for clarity. This line represents the entire function.
Since it's a linear function, it’s always a straight line. Each point on the line corresponds to a set of inputs and outputs as defined by the function. Keep in mind that graphing is a visual tool that allows us to easily interpret the characteristics of a function, such as its rate of change and intercepts.

Finding Zeros of Functions

Finding the zero of a function means identifying the input value that results in an output of zero, or where the graph of the function intersects the x-axis. For our function \( f(x) = x + 2 \), the zero can be found algebraically by setting the function equal to zero and solving for \( x \):

• Set \( f(x) = 0 \)
• \( x + 2 = 0 \)
• Solve for \( x \) by subtracting \(2\) from both sides: \( x = -2 \)

The zero, \( x = -2 \), was confirmed graphically when the line crossed the x-axis at this point.
Zeros are important because they provide valuable information about the function’s roots and its behavior in relation to the x-axis. Understanding how to find them is essential in solving many algebraic problems.