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Chapter 1: Problem 6

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

Short Answer

Expert verified
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 by step solution

01

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 \((0, -3)\).
02

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 \(\left(\frac{9}{4}, 0\right)\).
03

Determine the Slope

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope. Here, the slope \(m\) is \(\frac{4}{3}\).
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(9-(x+1)<0\) (b) \(9-(x+1) \geq 0\)

Use the intersection-of-graphs method to approximate each solution to the nearest hundredth. $$-0.15(6+\sqrt{2} x)+1.4(2 \pi x-6.1)=10$$

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If \(c\) is a zero of the linear function \(f(x)=m x+b, m \neq 0\), then the point at which the graph intersects the \(x\) -axis has coordinates (______,_______)
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