/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 For each of the functions in Exe... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 16

For each of the functions in Exercises 16-18, identify any horizontal intercepts and vertical asymptotes. Then, if possible, use technology to graph each function and verify your results. $$ f(x)=\frac{3 x-13}{x-4} $$

Short Answer

Expert verified
Horizontal intercept: \( \left( \frac{13}{3}, 0 \right) \). Vertical asymptote: \( x = 4 \).

Step by step solution

01

Find the horizontal intercept

The horizontal intercept occurs where the function equals zero, that is, where the numerator of \( f(x) = \frac{3x-13}{x-4} \) is zero. Set the numerator equal to zero and solve for \( x \):\[ 3x - 13 = 0 \Rightarrow 3x = 13 \Rightarrow x = \frac{13}{3} \]. Thus, the horizontal intercept is at \( \left( \frac{13}{3}, 0 \right) \).
02

Find the vertical asymptote

The vertical asymptote occurs where the denominator of \( f(x) \) is zero, that is, where \( x - 4 = 0 \). This gives \( x = 4 \). Thus, there is a vertical asymptote at \( x = 4 \).
03

Verify results using graphing technology

Use a graphing calculator or software to plot \( f(x) = \frac{3x-13}{x-4} \). Verify that the function crosses the x-axis at \( \left( \frac{13}{3}, 0 \right) \) and there is a vertical asymptote at \( x = 4 \). The graph should reflect these properties accurately.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

horizontal intercepts
A horizontal intercept, also known as an x-intercept, is where the function crosses the x-axis. For the function given, \( f(x)=\frac{3 x-13}{x-4} \), we find the horizontal intercept by setting the numerator equal to zero and solving for \( x \). This means we solve the equation \( 3x-13=0 \). When solving this equation, we get: \( 3x=13 \; \Rightarrow \; x=\frac{13}{3} \). So the horizontal intercept is at \( \left(\frac{13}{3}, 0 \right) \).

This means the function touches the x-axis at \( x=\frac{13}{3} \), and the y-coordinate is zero at this point. Knowing how to find the horizontal intercept helps in understanding how the function behaves and where it crosses the x-axis.
vertical asymptotes
Vertical asymptotes are lines that the graph of a function approaches but never touches. They often occur when the denominator of a rational function is zero because division by zero is undefined.

For the function \( f(x)=\frac{3 x-13}{x-4} \), we find the vertical asymptote by setting the denominator equal to zero and solving for \(x\). This results in the equation \( x-4=0 \), giving us \( x=4 \). Thus, the vertical asymptote is at \( x=4 \).

Graphically, this means that as \( x \) approaches 4 from either the left or the right, the value of \( f(x) \) increases or decreases without bound, creating a vertical line that the graph cannot cross or touch. Identifying vertical asymptotes is crucial for understanding how to plot the graph as it shows where the function's value 'blows up' to infinity.
graphing technology
To verify the results obtained analytically, we can use graphing technology such as graphing calculators or software. These tools allow us to visualize the function and ensure our intercepts and asymptotes are correctly calculated.

For the function \( f(x)=\frac{3 x-13}{x-4} \), when plotting it using graphing technology, you'll observe the graph intersects the x-axis at the horizontal intercept \( \left(\frac{13}{3}, 0 \right) \). Additionally, you'll see the graph approaching but never crossing the vertical line at \( x=4 \), which is our vertical asymptote.

Graphing tools can also help to see how the function behaves near these critical points and understand the overall shape. This visualization confirms the mathematical findings and serves as a practical check for correctness.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Transform the function \(f(x)=x^{2}\) into a new function \(g(x)\) by compressing \(f(x)\) by a factor of \(\frac{1}{4},\) then shifting the result horizontally left three units, and finally shifting it down by six units. Find the equation of \(g(x)\) and sketch it by hand.

(Graphing program optional) a. Write each of the following functions in both the \(a-b-c\) and the \(a-h-k\) forms. Is one form easier than the other for finding the vertex? The \(x\) - and \(y\) -intercepts? \(y_{1}=2 x^{2}-3 x-20 \quad y_{3}=3 x^{2}+6 x+3\) \(y_{2}=-2(x-1)^{2}-3 \quad y_{4}=-(2 x+4)(x-3)\) b. Find the vertex and \(x\) - and \(y\) -intercepts and construct a graph by hand for each function in part (a). If you have access to a graphing program, check your work.

Estimate the maximum number of horizontal intercepts for each of the polynomial functions. Then graph the function using technology to find the actual number. a. \(y=x^{4}-2 x^{2}-5\) b. \(y=4 t^{6}+t^{2}\) c. \(y=x^{3}-3 x^{2}+4\) d. \(y=5+x\)

The wind chill temperature is the apparent temperature caused by the extra cooling from the wind. A rule of thumb for estimating the wind chill temperature for an actual temperature \(t\) that is above \(0^{\circ}\) Fahrenheit is \(W(t)=t-1.5 S_{0}\), where \(S_{0}\) is any given wind speed in miles per hour. a. If the wind speed is 25 mph and the actual temperature is \(10^{\circ} \mathrm{F}\), what is the wind chill temperature? We know how to convert Celsius to Fahrenheit; that is, we can write \(t=F(x),\) where \(F(x)=32+\frac{9}{5} x,\) with \(x\) the number of degrees Celsius and \(F(x)\) the equivalent in degrees Fahrenheit. b. Construct a function that will give the wind chill temperature as a function of degrees Celsius. c. If the wind speed is \(40 \mathrm{mph}\) and the actual temperature is \(-10^{\circ} \mathrm{C},\) what is the wind chill temperature?

In Exercises \(16-22,\) show that the two functions are inverses of each other. $$ f(x)=\sqrt{x-1} \text { (where } x>1 \text { ) and } g(x)=x^{2}+1 \text { (where } \left.x>0\right) $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.