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

Verified step by step guidance

1. Step 1: Start by substituting the given expression (x - 6) for the variable r in the function f(r). The function is f(r) = √(r + 6) + 3, so replace r with (x - 6). This gives f(x - 6) = √((x - 6) + 6) + 3.
2. Step 2: Simplify the expression inside the square root. Combine like terms in (x - 6) + 6. This simplifies to x, so the function becomes f(x - 6) = √(x) + 3.
3. Step 3: Verify that the simplified function f(x - 6) = √(x) + 3 is valid for all x where the square root is defined. Recall that the square root function is only defined for non-negative values, so x must be greater than or equal to 0 (x ≥ 0).
4. Step 4: Conclude that the simplified function is f(x - 6) = √(x) + 3, and it is valid for x ≥ 0.
5. Step 5: If needed, you can now evaluate the function for specific values of x by substituting those values into the simplified expression f(x - 6) = √(x) + 3.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable in a function. In this case, we replace 'r' in the function f(r) = √(r + 6) + 3 with the expression (x - 6). This process allows us to determine the output of the function for that particular input.

Square Root Function

The square root function, denoted as √(x), is defined as the value that, when squared, gives x. It is important to understand that the square root function only produces non-negative outputs. In the context of the given function, we need to ensure that the expression inside the square root, (r + 6), remains non-negative for valid evaluations.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to understand or compute. After evaluating the function at the given input, we may need to combine like terms or reduce fractions. In this case, after substituting (x - 6) into the function, we will simplify the resulting expression to present the final answer clearly.