Let ƒ(x)=-3x+4 and g(x)=-x2+4x+1. Find each of the following. Simplify if necessary. g(-x)

Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 62

Chapter 3, Problem 62

Let ƒ(x)=-3x+4 and g(x)=-x²+4x+1. Find each of the following. Simplify if necessary. g(-x)

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Identify the given functions: $f(x) = -3x + 4$ and $g(x) = -x^{2} + 4x + 1$.

The problem asks to find $g(-x)$, which means we need to substitute $-x$ into the function $g(x)$ wherever $x$ appears.

Replace every $x$ in $g(x)$ with $-x$: $g(-x) = -(-x)^{2} + 4(-x) + 1$.

Simplify each term carefully: $(-x)^{2}$ becomes $x^{2}$ because squaring a negative number makes it positive; multiply $4$ by $-x$ to get $-4x$.

Write the simplified expression for $g(-x)$ as $-x^{2} - 4x + 1$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Notation and Evaluation

Function notation, such as f(x) or g(x), represents a rule that assigns each input x to an output. Evaluating a function means substituting a specific value or expression for x and simplifying the result. For example, g(-x) means replacing every x in g(x) with -x.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. In this problem, g(x) is a quadratic polynomial, which means it includes an x² term. Understanding how to manipulate and simplify polynomials is essential.

Simplification of Algebraic Expressions

Simplification involves combining like terms and applying arithmetic operations to rewrite expressions in a simpler or more standard form. When substituting -x into g(x), it is important to carefully apply exponent rules and combine terms to get the simplest expression.

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