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Chapter 3: Problem 22

Find the \(x\) - and \(y\) -intercepts of the rational function. $$s(x)=\frac{3 x}{x-5}$$

Short Answer

Expert verified
The x-intercept and y-intercept are both at (0, 0).

Step by step solution

01

Finding the x-intercept

The x-intercept of a function occurs when the output of the function is zero, i.e., \( s(x) = 0 \). Set the numerator of the rational function equal to zero because a fraction is zero when its numerator is zero. So, solve the equation \( 3x = 0 \) to find the x-value of the x-intercept.
02

Solving for x

Since \( 3x = 0 \), divide both sides by 3 to solve for \( x \).\[ x = \frac{0}{3} = 0 \]Thus, the x-intercept is at \( (0, 0) \).
03

Finding the y-intercept

The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) in the function \( s(x) = \frac{3x}{x-5} \) to find the y-value of the y-intercept.
04

Solving for y-intercept

Substitute \( x = 0 \) into the function:\[ s(0) = \frac{3(0)}{0-5} = \frac{0}{-5} = 0 \]Thus, the y-intercept is also at \( (0, 0) \).

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Key Concepts

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

x-intercept
The x-intercept of a rational function is where the graph crosses the x-axis. This point occurs when the output of the function, which is the y-value, is zero. For the function \( s(x) = \frac{3x}{x-5} \), finding the x-intercept is all about setting the entire fraction equal to zero.
Remember, a fraction is only zero when its numerator is zero. This is because if the numerator is zero, then dividing zero by any non-zero number results in zero. Consequently, solve the equation \( 3x = 0 \) to find the x-intercept.
  • Divide both sides of the equation by 3 to isolate \( x \).
  • The solution is \( x = 0 \).
So, the x-intercept is at the point \((0,0)\). This means the graph of \( s(x) \) crosses the x-axis at the origin.
y-intercept
The y-intercept of a function is the point at which the graph crosses the y-axis. This happens when \( x = 0 \). To find the y-intercept of a rational function like \( s(x) = \frac{3x}{x-5} \), simply substitute \( x = 0 \) into the function.

By plugging in zero for \( x \), the evaluation becomes:
  • \( s(0) = \frac{3(0)}{0-5} \)
  • This simplifies to \( \frac{0}{-5} = 0 \).
Thus, the y-intercept is also the point \((0, 0)\). This means the graph crosses the y-axis at the origin, emphasizing it is an important characteristic when sketching the function.
numerator and denominator analysis
Analyzing the numerator and denominator is crucial for understanding the behavior of rational functions. Let's break down how each affects the function \( s(x) = \frac{3x}{x-5} \).
Numerator Analysis:
  • The numerator is \( 3x \). It determines the x-intercepts of the function.
  • If \( 3x = 0 \), then \( x = 0 \), which gives us the x-intercept.
Denominator Analysis:
  • The denominator is \( x-5 \). It affects the domain of the function because division by zero is undefined.
  • Set the denominator equal to zero: \( x-5 = 0 \). Solving gives \( x=5 \), indicating a vertical asymptote at \( x=5 \).
These analyses tell us key features: the x-intercept at \((0, 0)\), the y-intercept at \((0, 0)\), and a vertical asymptote at \( x=5 \). These insights are instrumental in sketching and understanding the graph's shape and behavior.

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Most popular questions from this chapter

Graphing Quadratic Functions A quadratic function \(f\) is given. (a) Express \(f\) in standard form. (b) Find the vertex and \(x\) and \(y\) -intercepts of \(f .\) (c) Sketch a graph of \(f .\) (d) Find the domain and range of \(f\). $$f(x)=x^{2}+4 x+3$$

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