/*! 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 56 If \(f(x)=3 x^{2}-7\) and \(g(x)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 56

If \(f(x)=3 x^{2}-7\) and \(g(x)=2 x+a,\) find \(a\) so that the \(y\) -intercept of \(f \circ g\) is 68 .

Short Answer

Expert verified
a = \pm 5

Step by step solution

01

Understand the problem

We need to find the value of 'a' such that the y-intercept of the composite function \( f \circ g \) is 68. The y-intercept is found by evaluating the function at x = 0.
02

Compose the functions

The composite function \( f \circ g \) means applying g(x) first and then applying f to the result. This is written as \( f(g(x)) \). First, calculate \( g(0) \: g(0) = 2(0) + a = a \).
03

Apply f to g(0)

Use the result from the previous step to find \( ( f \circ g ) (0) = f(g(0)) \). This means we need to evaluate \( f(a) \): \( f(a) = 3a^2 - 7 \).
04

Set up the equation

The y-intercept is given as 68, so we set up the equation: \[ 3a^2 - 7 = 68 \]
05

Solve for a

Add 7 to both sides of the equation: \ 3a^2 = 75.\ Divide both sides by 3: \[ a^2 = 25 \]. Take the square root of both sides to get: \[ a = \pm 5 \]

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.

Function Composition
To understand function composition, think of it as a process where you combine two functions to create a new one. Here's how it works:
  • First, you apply the inner function.
  • Then, you apply the outer function to the result of the inner function.
In mathematical terms, if you have two functions, say \( f(x) \) and \( g(x) \), the composite function \( f \circ g \) is written as \( f(g(x)) \). This means you first find \( g(x) \) and then use that result as the input for \( f(x) \). For example:
  • If \( g(x) = 2x + a \).
  • And \( f(x) = 3x^{2} - 7 \).
  • Then, the composite function \( f \circ g \) becomes \( f(g(x)) = f(2x + a) \).
Understanding this allows you to solve more complex problems where functions are nested inside each other. In our exercise, we first find \( g(0) \) to get \( g(0) = a \). Then we substitute \( a \) into \( f \) to find the composite function's y-intercept.
Y-Intercept
Finding the y-intercept of a function is crucial because it tells you where the graph of the function crosses the y-axis. The y-intercept is the value of the function when \( x = 0 \). Here's how to find it:
  • Set \( x \) to 0 in the function.
  • Calculate the resulting value.
For our problem, we are looking for the y-intercept of the composite function \( f \circ g \). This means we need to find \( ( f \circ g )(0) \). We first find \( g(0) \). Given \( g(x) = 2x + a \), we have:
  • \( g(0) = 2(0) + a = a \).
Next, we use this \( a \) in \( f(x) = 3x^{2} - 7 \). So, we calculate \( f(a) \) to find the y-intercept:
  • \( f(a) = 3a^2 - 7 \).
From there, we set it equal to the y-intercept we want (68), creating the equation \( 3a^2 - 7 = 68 \).
Solving Quadratic Equations
Solving quadratic equations is a key skill in algebra. A quadratic equation is generally in the form \( ax^2 + bx + c = 0 \). There are various methods to solve it, such as factoring, completing the square, or using the quadratic formula. In our exercise, we have the quadratic equation \( 3a^2 - 7 = 68 \). To solve it:
  • Add 7 to both sides to move the constant term:
  • \( 3a^2 = 75 \).
  • Divide both sides by 3 to isolate \( a^2 \):
  • \( a^2 = 25 \).
  • Take the square root of both sides to solve for \( a \):
  • \( a = \pm 5 \).
The solution tells us that \( a \) can be either 5 or -5. Both values satisfy the condition that the y-intercept of the composite function \( f \circ g \) is 68. Quadratic equations often have two solutions, so it's important to consider both.

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

Graph each function using a graphing utility and the Change-of-Base Formula. \(y=\log _{5} x\)

Solve each equation. $$ \log _{5}\left(x^{2}+x+4\right)=2 $$

If the average annual inflation rate is \(3.1 \%\), how long will it take for the CPI index to double?

Graph each function. Based on the graph, state the domain and the range, and find any intercepts. $$ f(x)=\left\\{\begin{array}{ll} \ln (-x) & \text { if } x \leq-1 \\ -\ln (-x) & \text { if }-1

Graph each function. Based on the graph, state the domain and the range, and find any intercepts. $$ f(x)=\left\\{\begin{array}{ll} -\ln x & \text { if } 0
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Pure 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.