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

Verified step by step guidance

1. Identify the function ƒ(x) which is given as -3x + 4.
2. Substitute 'a + 4' in place of 'x' in the function ƒ(x). This means wherever you see 'x', replace it with 'a + 4'.
3. Apply the substitution: ƒ(a + 4) = -3(a + 4) + 4.
4. Distribute the -3 across the terms inside the parentheses: -3(a + 4) = -3a - 12.
5. Combine the constant terms: -3a - 12 + 4 = -3a - 8.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. For example, if you have a function f(x) and you want to find f(a+4), you replace x with (a+4) in the function's expression. This process is fundamental in algebra as it allows us to compute values based on defined relationships.

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In the given function f(x) = -3x + 4, the slope is -3, indicating a decrease in value as x increases. Understanding linear functions is crucial for analyzing their behavior and graphing them.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to work with. This can include combining like terms, factoring, or reducing fractions. In the context of evaluating functions, simplification helps clarify the output and ensures that the final answer is presented in the most concise manner possible.