Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. g(-x)

Verified step by step guidance

1. Identify the function g(x) which is given as g(x) = -x^2 + 4x + 1.
2. Replace every instance of x in the function g(x) with -x to find g(-x).
3. Substitute -x into the function: g(-x) = -(-x)^2 + 4(-x) + 1.
4. Simplify the expression by squaring -x and distributing the 4: g(-x) = -x^2 - 4x + 1.
5. The expression for g(-x) is now simplified to g(-x) = -x^2 - 4x + 1.

Key Concepts

Function Notation

Function notation is a way to represent mathematical functions using symbols. In this context, ƒ(x) and g(x) denote two different functions, where 'x' is the input variable. Understanding how to read and interpret function notation is essential for evaluating functions at specific values, such as g(-x).

Evaluating Functions

Evaluating a function involves substituting a specific value for the variable in the function's expression. For example, to find g(-x), you replace 'x' in the function g(x) with '-x'. This process is crucial for determining the output of the function based on the given input, which is a fundamental skill in algebra.

Simplifying Expressions

Simplifying expressions is the process of reducing a mathematical expression to its simplest form. This may involve combining like terms, factoring, or applying algebraic identities. In the context of finding g(-x), simplifying the resulting expression is important to present the final answer clearly and concisely.