/*! 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 6 Let \(f(x)=2 x^{2}+x-4\) and \(g... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Let \(f(x)=2 x^{2}+x-4\) and \(g(x)=3-x^{2} .\) Find the specified values. $$ g(f(0)) $$

Short Answer

Expert verified
g(f(0)) = -13

Step by step solution

01

Evaluate f(0)

To find \( f(0) \), substitute \( x = 0 \) into the function \( f(x) = 2x^2 + x - 4 \). This gives us: \[ f(0) = 2(0)^2 + 0 - 4 = -4 \] Thus, \( f(0) = -4 \).
02

Evaluate g(f(0))

Now that we know \( f(0) = -4 \), we need to find \( g(f(0)) = g(-4) \). Substitute \( x = -4 \) into the function \( g(x) = 3 - x^2 \). This gives us: \[ g(-4) = 3 - (-4)^2 = 3 - 16 = -13 \] Thus, \( g(f(0)) = -13 \).

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.

Quadratic Functions
Quadratic functions are a type of polynomial function, distinguished by their highest degree of 2. When graphed, they form a distinctive parabolic shape. A quadratic function generally comes in the form of \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \).
Quadratics can have various forms:
  • Standard form: \( ax^2 + bx + c \)
  • Vertex form: \( a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.
  • Factored form: \( a(x-p)(x-q) \), where \( p \) and \( q \) are the roots or zeros of the quadratic.
Quadratic functions model many natural phenomena and solve real-world problems.
They often represent scenarios involving projectile motion, optimization problems, and area calculations.
Function Evaluation
Function evaluation entails substituting a given input into a function to find its corresponding output. It's like plugging numbers into a machine and getting an answer.
If you have a function \( f(x) = 2x^2 + x - 4 \) and want to evaluate \( f(0) \), you simply replace every \( x \) with 0.
The calculation would look like:
  • First, substitute: \( f(0) = 2(0)^2 + 0 - 4 \)
  • Next, simplify: \( f(0) = 0 + 0 - 4 \)
  • Result: \( f(0) = -4 \)
This process is repeated whenever you need to evaluate a function at different points, providing valuable insight into how the function behaves.
It's essential when working with any mathematical function because it tells you the output (or value) of a function at a particular input.
Nested Functions
Nested functions involve placing one function inside another. This is often referred to as function composition.
In mathematics, it appears as \( g(f(x)) \), which indicates that you're using the output from \( f(x) \) as the input for \( g(x) \).
Let's break it down with an example:
  • Suppose \( f(x) = 2x^2 + x - 4 \) and \( g(x) = 3 - x^2 \).
  • First, find \( f(0) \): substitute 0 in \( f(x) \) to get \( -4 \).
  • Next, find \( g(f(0)) \) which becomes \( g(-4) \).
  • By substituting \( -4 \) into \( g(x) \): \( g(-4) = 3 - (-4)^2 \).
  • Simplify: \( g(-4) = 3 - 16 = -13 \).
Function composition, such as nested functions, points to an important concept in mathematics where you 'stack' functions to create complex operations.
This method is useful for breaking down functions into manageable parts and solving layered problems.

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

Solve the equation. $$ |x|=\pi $$

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: 3 x-y=0 ; P=(1,3)\)

A single respiratory cycle includes one inhalation and one exhalation. During one respiratory cycle of a certain person at rest, the rate of flow \(R\) (in liters per second) of air into a person's lungs at time \(t\) (in seconds) is given by $$ R=0.5 \sin \frac{2 \pi}{5} t $$ a. How long does it take to complete one respiratory cycle? b. How many respiratory cycles are completed in one minute? c. Graph one complete cycle, starting at time \(t=0\). d. Interpret the meaning of positive and negative values of \(R\) e. To the nearest hundredth, find \(R\) when \(t=3\) seconds.

Modify the expression, and then find its approximate value by calculator or computer. $$ \frac{(0.123)^{9000}}{(0.125)^{9000}} $$

Evaluate the expression. $$ |-\sqrt{2}|^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

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.