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

Verified step by step guidance

1. Identify the function ƒ(x) which is given as -3x + 4.
2. Replace every instance of x in the function ƒ(x) with -x to find ƒ(-x).
3. Substitute -x into the function: ƒ(-x) = -3(-x) + 4.
4. Simplify the expression by distributing and combining like terms.
5. The simplified form of ƒ(-x) is the result of the above simplification.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this case, evaluating ƒ(-x) means replacing every instance of 'x' in the function ƒ(x) with '-x'. This process is fundamental in understanding how functions behave under transformations and is crucial for solving the given problem.

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form ƒ(x) = mx + b, where m is the slope and b is the y-intercept. The function ƒ(x) = -3x + 4 is linear, indicating that its graph is a straight line. Understanding the properties of linear functions, such as slope and intercepts, is essential for analyzing their behavior and transformations.

Algebraic Simplification

Algebraic simplification is the process of rewriting an expression in a more concise or manageable form. This may involve combining like terms, factoring, or reducing fractions. In the context of the question, simplifying the result of ƒ(-x) is necessary to present the final answer clearly and effectively, ensuring that it is in its simplest form.