In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5 b. f(x + 1)

Verified step by step guidance

1. Start by identifying the given function: \( f(x) = 4x + 5 \).
2. To evaluate \( f(x + 1) \), substitute \( x + 1 \) into the function in place of \( x \).
3. Replace \( x \) with \( x + 1 \) in the expression: \( f(x + 1) = 4(x + 1) + 5 \).
4. Distribute the 4 across the terms inside the parentheses: \( 4(x + 1) = 4x + 4 \).
5. Combine like terms: \( 4x + 4 + 5 \) simplifies to \( 4x + 9 \).

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable in a function. In this case, to evaluate f(x + 1), we replace x in the function f(x) = 4x + 5 with (x + 1). This process allows us to determine the output of the function for a given input.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. After substituting (x + 1) into the function, we will perform operations such as distribution and combining like terms to simplify the resulting expression. This step is crucial for presenting the final answer clearly.

Linear Functions

A linear function is a polynomial function of degree one, which can be expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The function f(x) = 4x + 5 is linear, indicating that its graph is a straight line. Understanding the properties of linear functions helps in visualizing and interpreting the results of function evaluations.