/*! 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 108 Write an equation of a function ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 108

Write an equation of a function that meets the given conditions. Answers may vary. \(x\) -intercepts: (4,0) and (2,0) vertical asymptote: \(x=1\) horizontal asymptote: \(y=1\) \(y\) -intercept: (0,8)

Short Answer

Expert verified
f(x) = \frac{(x - 4)(x - 2)}{x - 1}

Step by step solution

01

Identify the General Form of the Function

Given the intercepts and asymptotes, a rational function is suitable. The general form can be written as: \[ f(x) = \frac{(x - a)(x - b)}{(x - c)} \] We will modify this to fit all conditions.
02

Apply the x-intercepts

The x-intercepts are given as (4,0) and (2,0). This leads to the numerator being of the form: \[ (x - 4)(x - 2) \]
03

Incorporate the Vertical Asymptote

The vertical asymptote is specified as \(x=1\). Hence, the denominator should be \(x-1\), making our function: \[ f(x) = \frac{(x - 4)(x - 2)}{x - 1} \]
04

Horizontal Asymptote Adjustment

The horizontal asymptote is given as \(y=1\). For the horizontal asymptote of a rational function to be 1, we need the degrees of the numerator and denominator to be equal and their leading coefficients to form their ratio as 1. Hence, multiply by the leading coefficient of 1 which is already fulfilled: \[ f(x) = \frac{(x - 4)(x - 2)}{x - 1} \]
05

Check the y-intercept

Substitute \(x = 0\) to check if the y-intercept is 8 and adjust the coefficient as necessary: \[ f(0) = \frac{(0 - 4)(0 - 2)}{0 - 1} = \frac{8}{1} = 8 \]Thus our function is correct.

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.

x-intercepts
The x-intercepts are where the graph of the function crosses the x-axis. These are the points where the output value (y) is zero. To find the x-intercepts, we set the numerator of the rational function to zero. In this problem, the x-intercepts are given as (4,0) and (2,0). This means the function will have factors \((x - 4) \) and \((x - 2) \) in the numerator. When the numerator equals zero, the entire fraction equals zero. Therefore, the equation \[ f(x) = \frac{(x - 4)(x - 2)}{...} \] will give us the x-intercepts of (4,0) and (2,0). This is a crucial step in constructing the function.
vertical asymptote
Vertical asymptotes occur where the denominator equals zero but the numerator does not. They represent x-values where the function tends to infinity. For a vertical asymptote at x = 1, the denominator of the rational function must have a factor of \((x - 1) \). This makes our function look like: \[ f(x) = \frac{(x - 4)(x - 2)}{x - 1} \]. As x approaches 1 from either side, the value of the function increases or decreases without bound, which characterizes a vertical asymptote. This behavior is fundamental to understanding the nature of rational functions.
horizontal asymptote
Horizontal asymptotes provide information on the function's end behavior, indicating the value that the function approaches as x tends to positive or negative infinity. For a horizontal asymptote at y = 1, the degrees of the numerator and the denominator must be the same, and their leading coefficients鈥� ratio must be 1. In our function \[ f(x) = \frac{(x - 4)(x - 2)}{x - 1}, \] the degrees of both the numerator and the denominator match (both are 2 in simplified form). The leading coefficients are both effectively 1, ensuring our function approaches y = 1 as x approaches infinity. This aspect helps in providing a global picture of the function's graph.
y-intercept
The y-intercept is where the graph crosses the y-axis, meaning x equals zero. To find the y-intercept, substitute x = 0 into the function and solve for y. This gives us the coordinate (0, y). For the function \[ f(x) = \frac{(x - 4)(x - 2)}{x - 1}, f(0) = \frac{(0 - 4)(0 - 2)}{0 - 1} = \frac{8}{1} = 8. \] Therefore, the y-intercept is (0,8), matching the given conditions. Verifying this intercept ensures our function is correctly aligned with all given constraints, completing our function construction.

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

A professional fireworks team shoots an 8 -in. mortar straight upwards from ground level with an initial velocity of \(216 \mathrm{ft} / \mathrm{sec} .\) (See Example 6\()\) a. Write a function modeling the vertical position \(s(t)\) (in \(\mathrm{ft}\) ) of the shell at a time \(t\) seconds after launch. b. The mortar is designed to explode when the shell is at its maximum height. How long after launch will the shell explode? (Hint: Consider the vertex formula from Section 3.1.) c. The spectators can see the shell rising once it clears a 200 -ft tree line. For what period of time after launch is the shell visible before it explodes?

Suppose that \(y\) varies inversely as the cube of \(x\). If the value of \(x\) is decreased to \(\frac{1}{4}\) of its original value, what is the effect on \(y\) ?

Simplify the expression. $$ \frac{15 x^{3}}{5 x} $$

The average round trip speed \(S\) (in mph) of a vehicle traveling a distance of \(d\) miles in each direction is given by $$S=\frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}}$$ where \(r_{1}\) and \(r_{2}\) are the rates of speed for the initial trip and the return trip, respectively. a. Suppose that a motorist travels \(200 \mathrm{mi}\) from her home to an athletic event and averages \(50 \mathrm{mph}\) for the trip to the event. Determine the speeds necessary if the motorist wants the average speed for the round trip to be at least \(60 \mathrm{mph}\) ? b. Would the motorist be traveling within the speed limit of \(70 \mathrm{mph} ?\)

The light from a lightbulb radiates outward in all directions. a. Consider the interior of an imaginary sphere on which the light shines. The surface area of the sphere is directly proportional to the square of the radius. If the surface area of a sphere with a \(10-\mathrm{m}\) radius is \(400 \pi \mathrm{m}^{2}\), determine the surface area of a sphere with a \(20-\mathrm{m}\) radius. b. Explain how the surface area changed when the radius of the sphere increased from \(10 \mathrm{~m}\) to \(20 \mathrm{~m}\). c. Based on your answer from part (b) how would you expect the intensity of light to change from a point \(10 \mathrm{~m}\) from the lightbulb to a point \(20 \mathrm{~m}\) from the lightbulb? d. The intensity of light from a light source varies inversely as the square of the distance from the source. If the intensity of a lightbulb is 200 lumen/m \(^{2}\) (lux) at a distance of \(10 \mathrm{~m}\), determine the intensity at \(20 \mathrm{~m}\).
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.