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Chapter 5: Problem 24

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. \((f+h)(-2)\)

Short Answer

Expert verified
\((f + h)(-2) = -10\)

Step by step solution

01

Understand the Problem

The task is to find the value of \(f + h\) at the input \(-2\). This requires adding the functions \(f(x)\) and \(h(x)\) first and then evaluating the resulting function at \(-2\).
02

Add the Functions

The function \(f(x) = x^2 - 9\) and \(h(x) = x - 3\). To find \( (f + h)(x) \), add \(f(x)\) and \(h(x)\) together: \( (f + h)(x) = f(x) + h(x) = (x^2 - 9) + (x - 3) = x^2 + x - 12 \).
03

Evaluate at \x = -2\

Substitute \x = -2\ into the combined function \( (f + h)(x) = x^2 + x - 12 \): \( (f + h)(-2) = (-2)^2 + (-2) - 12 = 4 - 2 - 12 = -10 \).

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Key Concepts

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

function evaluation
Function evaluation is the process of determining the output of a function for a certain input value. Imagine a function like a machine: you insert an input and get out an output.

Given a function, say, \( g(x) = 2x \), to evaluate \( g \) at an input of \( x = 3 \), you simply substitute 3 into the function:

\[ g(3) = 2(3) = 6. \]

This means that when the input is 3, the output of the function \( g(x) \) is 6.
polynomial functions
Polynomial functions are algebraic expressions that include terms in the form \( ax^n \), where \( a \) is a coefficient, \( x \) is a variable, and \( n \) is a non-negative integer. These functions can be simple or complex depending on the number of terms.

For example, \( f(x) = x^2 - 9 \) is a polynomial function. It has two terms: \( x^2 \) and \( -9 \). Its degree is 2 because the highest power of \( x \) is 2.

Polynomials can have many terms:
  • A monomial has one term (e.g., \( 3x \)).
  • A binomial has two terms (e.g., \( x^2 - 5 \)).
  • A trinomial has three terms (e.g., \( x^3 + 2x - 1 \)).
Understanding these basics helps you manipulate and solve polynomial functions.
algebraic expressions
Algebraic expressions are mathematical phrases combining numbers, variables, and operations like addition and multiplication. They form the foundation for various types of functions, including polynomial functions.

For example, \( 2x + 3 \) is an algebraic expression that combines the variable \( x \) with numbers using addition. When you work with algebraic expressions, you will often:
  • Add or subtract expressions.
  • Multiply or divide expressions.
  • Combine like terms.
Understanding how to manipulate these expressions allows you to perform function addition, among other operations.

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Most popular questions from this chapter

Write each number in scientific notation. $$ 0.875 $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ a^{-3} a^{2} a^{-4} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \left(\frac{-3 x^{4} y^{6}}{15 x^{-6} y^{7}}\right)^{-3} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \left(\frac{5 m^{4} n^{-3}}{m^{-5} n^{2}}\right)^{-2} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{4 a^{5}\left(a^{-1}\right)^{3}}{\left(a^{-2}\right)^{-2}} $$
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