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Chapter 7: Problem 14

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. $$ g(-x) $$

Short Answer

Expert verified
g(-x) = -x^2 - 4x + 1

Step by step solution

01

Understand the Function g(x)

The given function is \( g(x) = -x^2 + 4x + 1 \).
02

Substitute -x into g(x)

Replace every instance of \(x\) in \(g(x)\) with \(-x\). This gives us \(g(-x) = -(-x)^2 + 4(-x) + 1\).
03

Simplify the Expression

Simplify the expression: \( g(-x) = -(-x)^2 + 4(-x) + 1 = -x^2 - 4x + 1 \).

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Key Concepts

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

Polynomial Functions
Polynomial functions are mathematical expressions involving sums of powers of variables with coefficients. They are written in the form: oat P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0.oFor example, in the function provided, g(x) = -x^2 + 4x + 1,o -x^2 is the quadratic term with a coefficient of -1,o 4x is the linear term with a coefficient of 4,o and 1 is the constant term.oPolynomial functions can describe a wide range of behaviors and are essential in algebra and calculus.
Substitution in Functions
Substitution in functions involves replacing the variable with a different expression or value. This concept is crucial for evaluating functions at specific points or transforming functions for various analyses.

In the given problem, we need to find g(-x). To do this, we replace every occurrence of x in g(x) with -x.oOriginal function: g(x) = -x^2 + 4x + 1.oPerform substitution: g(-x) = -(-x)^2 + 4(-x) + 1.oCarefully substituting the values help avoid errors and aids in understanding transformations of functions.
Simplifying Expressions
Simplifying expressions is the process of reducing them to their simplest form. This involves combining like terms, applying arithmetic rules, and making the expression easier to work with.

Let's simplify the substituted function g(-x):oStarting with: g(-x) = -(-x)^2 + 4(-x) + 1oFirst, handle the exponentiation and signs: -(-x)^2 becomes -x^2 (since (-x)^2 is x^2)oNext, multiply the terms: 4(-x) becomes -4xoFinally, combine all the terms: g(-x) = -x^2 - 4x + 1.oSimplifying expressions correctly is key for solving many algebraic problems and ensures clarity in mathematical communication.

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Most popular questions from this chapter

For each function, find (a) \(f(2)\) and \((b) f(-1) .\) See Examples 4 and \(5 .\) $$ f=\\{(2,5),(3,9),(-1,11),(5,3)\\} $$

Find an equation of the line that satisfies the given conditions. See Example 4. Through \((2,-7) ;\) horizontal

Find the midpoint of each segment with the given endpoints. $$\left(-\frac{1}{3}, \frac{2}{7}\right) \text { and }\left(-\frac{1}{2}, \frac{1}{14}\right)$$

For each pair of functions, find \((a)(f+g)(x)\) and \((b)(f-g)(x) .\) See Example 2. $$ f(x)=4 x^{2}+8 x-3, g(x)=-5 x^{2}+4 x-9 $$

For Exercises \(47-58,(a)\) find the slope of the line through each pair of points, if possible, and (b) based on the slope, indicate whether the line through the points rises from left to right, falls from left to right, is horizontal, or is vertical. See Example 6 and FlGURE \(19 .\) $$(-2,2) \text { and }(4,-1)$$
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