Problem 32 Let $f(x)=x^{2}$ and \(g(x)=x-... [FREE SOLUTION]

91影视

Chapter 7: Problem 32

Let $f(x)=x^{2}$ and $g(x)=x-3 .$ Find each value or expression. $$ (f \circ g)(-2) $$

Short Answer

Expert verified

The value of $(f \circ g)(-2)$ is 25.

Step by step solution

Understand the Functions

The given functions are $f(x)=x^{2}$ and $g(x)=x-3$. These are straightforward polynomial functions.

Substitute g into f

We need to find $(f \circ g)(-2)$, which means plugging g(-2) into the f function. First, calculate $g(-2)$: Just substitute -2 into $g(x)$, which gives us $g(-2)=-2-3=-5$.

Compute the Result

Now that we have found g(-2) to be -5, we substitute it into the f function. That gives us $f(g(-2))=f(-5)=(-5)^{2}=25$.

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.

Polynomial Functions

Polynomial functions are a type of mathematical expression that involve variables raised to whole-number exponents and coefficients that are real numbers. They are essential in various areas of math because they offer simple models expressing real-world curves and behaviors. A polynomial function is usually in the form of:

A constant term which is a number without a variable (e.g., 3).
Linear terms, which are variables with an exponent of one (e.g., $2x$).
Quadratic terms, which are variables squared (e.g., $x^2$).
Cubic terms, which have a variable raised to the third power (e.g., $x^3$).

Consider the polynomial function $f(x) = x^2$. It is a quadratic function, meaning its highest exponent is 2, forming a parabola when graphed. In the given exercise, both functions $f(x) = x^2$ and $g(x) = x - 3$ are polynomial functions, with $f(x)$ being quadratic and $g(x)$ being linear.

Composite Functions

Composite functions involve combining two functions, where the output of one function becomes the input of another. To denote a composite function, we use the symbol $\circ$, read as "of" or "compose". Thus,

$(f \circ g)(x)$ means the function $g(x)$ is inside the function $f(x)$.
In practical terms, you first apply $g(x)$ and then use its result as the argument for $f(x)$.

For instance, in our given problem of $(f \circ g)(-2)$, we start by calculating $g(-2)$, and then take this value and substitute it into $f$. This creates a new composite function based on two simpler functions. Exploring composition helps deepen your understanding of how functions interact, leading to more complex expressions.

Function Evaluation

Function evaluation is the process of determining the output of a function given an input. It simply involves substituting a specific value into a function. This is crucial because it allows us to quantify and analyze relationships and changes between variables. Let's break it down:

To evaluate $g(x)$ at $x = -2$, we replace $x$ with $-2$, which results in: \[g(-2) = -2 - 3 = -5\].
Next, we take $g(-2) = -5$ and substitute it into $f(x)$, giving us: \[f(g(-2)) = f(-5) = (-5)^2 = 25\].

The purpose of this evaluation process is to find specific values of composite functions accurately. By evaluating correctly, students can check their understanding and ensure their ability to perform operations on functions effectively.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (f \circ g)(-2) $$

Short Answer

Step by step solution

Understand the Functions

Substitute g into f

Compute the Result

Key Concepts

Polynomial Functions

Composite Functions

Function Evaluation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Logic and Functions

Statistics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.