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. Step 1: Start with the given function f(x) = 4x + 5. This is a linear function where 'x' is the independent variable.
2. Step 2: Substitute (x + 1) into the function in place of 'x'. This means replacing every occurrence of 'x' in the function with (x + 1). The new expression becomes f(x + 1) = 4(x + 1) + 5.
3. Step 3: Apply the distributive property to simplify 4(x + 1). Multiply 4 by both 'x' and '1', resulting in 4x + 4.
4. Step 4: Combine like terms. Add the constant terms 4 and 5 together to simplify the expression further. This gives f(x + 1) = 4x + 9.
5. Step 5: The simplified expression for f(x + 1) is now f(x + 1) = 4x + 9. This is the final simplified form of the function evaluated at (x + 1).

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), you replace x in the function f(x) = 4x + 5 with (x + 1). This process allows you to find the output of the function for a new input.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. After substituting x + 1 into the function, you will need to simplify the resulting expression, combining like terms and performing any necessary arithmetic operations to achieve a clearer representation of the function's output.

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 is essential for evaluating and interpreting their behavior.