Evaluate each function at the given values of the independent variable and simplify. $f\left(r\right)=\sqrt{r+6}+3$
$\text{a. }f(x-6)$

Verified step by step guidance

1. Identify the given function: \(f(r) = \sqrt{r + 6} + 3\).
2. Understand that the problem asks to evaluate the function at \(x - 6\), which means we need to find \(f(x - 6)\) by substituting \(r\) with \(x - 6\) in the function.
3. Substitute \(r\) with \(x - 6\) in the function: \(f(x - 6) = \sqrt{(x - 6) + 6} + 3\).
4. Simplify the expression inside the square root: \((x - 6) + 6 = x\).
5. Write the simplified function: \(f(x - 6) = \sqrt{x} + 3\).

Key Concepts

Function Evaluation

Function evaluation involves substituting a given value or expression for the independent variable in the function's formula. This process helps determine the output of the function for specific inputs, such as replacing x with (x - 6) in f(x).

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form by performing arithmetic operations, combining like terms, and applying algebraic rules. This step is essential after substitution to present the function's value clearly and concisely.

Square Root Function

The square root function, denoted √, returns the non-negative root of a number or expression. Understanding its domain is important because the expression inside the root must be non-negative to yield real values, affecting the valid inputs for the function.