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Chapter 9: Problem 22

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1\) \(g(-x)\)

Short Answer

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

Step by step solution

01

- Substitute \( -x \) into the function \( g(x) \)

First, identify the function \( g(x) \). It is given by \( g(x) = -x^{2} + 4x + 1 \). To find \( g(-x) \), substitute \( -x \) for \( x \) in the function.
02

- Perform the substitution

Substitute \( -x \) into the function: \( g(-x) = -(-x)^{2} + 4(-x) + 1 \).
03

- Simplify the expression

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

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Key Concepts

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

Understanding Function Substitution
Let鈥檚 dive into the concept of function substitution. This means replacing the variable in the function with another expression. In this problem, the function is given by \( g(x) = -x^{2} + 4x + 1 \). Here, we are asked to find \( g(-x) \), which involves substituting \( -x \) for \( x \) in the function.
Literally, wherever there is an \( x \) in \( g(x) \), you replace it with \( -x \).
馃搶 This is crucial because it changes the behavior of the function. Let鈥檚 see the steps:
  • Original function: \( g(x) = -x^{2} + 4x + 1 \)
  • Substitute: wherever you see \( x \), replace it with \( -x \)
  • After substitution: \( g(-x) = -(-x)^{2} + 4(-x) + 1 \)
This transformation is called function substitution. It鈥檚 the backbone of transforming functions, and helps in understanding how functions change when inputs are modified.
Exploring Polynomial Functions
Polynomial functions are expressions consisting of variables and coefficients that are combined using only addition, subtraction, multiplication, and non-negative integer exponents. In our example, \( g(x) \) is a polynomial function:
  • \( g(x) = -x^{2} + 4x + 1 \)
  • The highest exponent of \( x \) is 2, defining it as a quadratic polynomial.
  • The terms are combined using addition and multiplication, which are key characteristics of polynomial functions.
Understanding polynomial functions is important because they form the foundation of many algebraic concepts and real-world applications. Knowing how to work with them, including their transformation, is essential for solving complex problems.
Simplification after Substitution
After substituting the function, the next step is simplification. Simplification means to make an expression easier to work with by combining like terms and performing basic arithmetic operations. Let's go through this step-by-step.
Starting from:
\( g(-x) = -(-x)^{2} + 4(-x) + 1 \)
  • First, simplify \( (-x)^{2} \); since squaring a negative makes it positive, it becomes \( x^{2} \).
  • Thus, \( -(-x)^{2} \) becomes \( -x^{2} \).

Next, combine the terms:
\( -x^{2} \)
\( 4(-x) = -4x \)
\( +1 \)
Putting it all together:
Expression: \( g(-x) = -x^{2} - 4x + 1 \)
Simplification makes the function easier to interpret and solve, eliminating complexity and ensuring clarity in mathematical expressions. 讛诪讬 This skill is fundamental in algebra and higher-level mathematics.

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