In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(r) = √(r + 6) +3 a. f(-6)

Verified step by step guidance

1. Step 1: Understand the problem. The function f(r) = √(r + 6) + 3 is given, and you are tasked with evaluating it at r = -6. This means substituting -6 for r in the function.
2. Step 2: Substitute r = -6 into the function. Replace r in the expression √(r + 6) + 3 with -6, resulting in √((-6) + 6) + 3.
3. Step 3: Simplify the expression inside the square root. Calculate (-6) + 6, which equals 0. The function now becomes √(0) + 3.
4. Step 4: Evaluate the square root. The square root of 0 is 0, so the expression simplifies further to 0 + 3.
5. Step 5: Add the remaining terms. Add 0 and 3 to complete the simplification, resulting in the final value of the function at r = -6.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable into a function to determine its output. In this case, we replace 'r' in the function f(r) = √(r + 6) + 3 with -6 to find f(-6). This process is fundamental in understanding how functions operate and how to compute their values.

Square Root Function

The square root function, denoted as √x, returns the non-negative value whose square equals x. In the context of the given function, √(r + 6) requires that the expression inside the square root, r + 6, must be non-negative for real number outputs. Understanding the domain of the square root function is crucial for evaluating it correctly.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to understand and work with. After evaluating the function at a specific value, such as f(-6), we may need to simplify the resulting expression, which can include combining like terms or reducing fractions. This skill is essential for clear mathematical communication and problem-solving.