Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Graph each linear function. Give the (a) \(x\) -intercept, (b) \(y\) -intercept. (c) domain, (d) range, and (e) slope of the line. $$f(x)=\frac{4}{3} x-3$$

Step-by-Step Solution

Answer

x-intercept: \(\left(\frac{9}{4}, 0\right)\), y-intercept: \((0, -3)\), slope: \(\frac{4}{3}\), domain: all real numbers, range: all real numbers.

Step 1: Determine the y-intercept

To find the y-intercept, set \(x = 0\) in the function. The y-intercept is the value of \(f(x)\) when \(x = 0\). Substitute to get \(f(0) = \frac{4}{3}(0) - 3 = -3\). Thus, the y-intercept is \((0, -3)\).

Step 2: Determine the x-intercept

To find the x-intercept, set \(f(x) = 0\) and solve for \(x\). This occurs when \(\frac{4}{3}x - 3 = 0\). Add 3 to both sides: \(\frac{4}{3}x = 3\). Multiply both sides by \(\frac{3}{4}\): \(x = \frac{9}{4}\). Therefore, the x-intercept is \(\left(\frac{9}{4}, 0\right)\).

Step 3: Determine the Slope

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope. Here, the slope \(m\) is \(\frac{4}{3}\).

Step 4: Determine the Domain and Range

For a linear function like \(f(x) = \frac{4}{3}x - 3\), the domain is all real numbers \((-abla, abla)\), as you can input any real number for \(x\). Similarly, the range is all real numbers \((-abla, abla)\), as the function produces any real value as output.

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear function is the point where the graph crosses the x-axis. At this point, the value of the function, or \( f(x) \), is zero. In essence, it is where the output of the function equals zero.

To find the x-intercept of the function \( f(x) = \frac{4}{3}x - 3 \), we set \( f(x) = 0 \).
Here's a step-by-step approach:

• Set up the equation: \( \frac{4}{3}x - 3 = 0 \).
• Add 3 to both sides to isolate the term with \( x \): \( \frac{4}{3}x = 3 \).
• Multiply both sides by the reciprocal of \( \frac{4}{3} \), which is \( \frac{3}{4} \), to solve for \( x \): \( x = \frac{9}{4} \).

Thus, the x-intercept is at the point \( \left(\frac{9}{4}, 0\right) \). It tells us that when \( x = \frac{9}{4} \), the function value is zero.

Exploring the Y-Intercept

The y-intercept is the point where the graph of a function crosses the y-axis. At this point, the value of \( x \) is zero. The y-intercept reveals the starting value of a function \( f(x) \) when you input \( x = 0 \).

For the function \( f(x) = \frac{4}{3}x - 3 \), finding the y-intercept involves the following:

• Set \( x = 0 \): This alters the function to \( f(0) = \frac{4}{3}(0) - 3 \).
• Calculate the expression: \( f(0) = -3 \).

This means the y-intercept of the function is at \( (0, -3) \). Hence, the graph will cross the y-axis at this point.

The Concept of Slope

The slope of a linear function represents its rate of change, indicating how steep the line is. Slope is essentially a measure of how much \( y \) changes with a change in \( x \).

In the slope-intercept form, \( y = mx + b \), \( m \) is the slope. For our function \( f(x) = \frac{4}{3}x - 3 \), the slope \( m \) is \( \frac{4}{3} \).
This can be interpreted as:

• For every increase of 1 in \( x \), \( y \) increases by \( \frac{4}{3} \).
• The line rises upwards from left to right, showing a positive slope.

Understanding slope helps us predict the behavior and direction of the line on the graph.

Deciphering the Domain

The domain of a function refers to all possible input values (\( x \) values) for which the function is defined.

In the case of the linear function \( f(x) = \frac{4}{3}x - 3 \), the domain is optioned as follows:

• Linear functions like this can accept any real number for \( x \).

Typically, the domain of a linear function is all real numbers, expressed as \( (-\infty, \infty) \). This implies that there are no restrictions on the input \( x \), and any real number can work.

Interpreting the Range

The range of a function is the set of all possible output values (\( y \) values) that it can produce. Just as with the domain, we consider what outputs are possible given the inputs.

For the function \( f(x) = \frac{4}{3}x - 3 \), its range is:

• For any real value of \( x \), there will be a corresponding real value for \( y \).

Thus, the range of a linear function is also all real numbers, expressed as \( (-\infty, \infty) \). This indicates that no matter the value of \( x \), the function will produce a real number output.