Question

Evaluate the function f left parenthesis r right parenthesis equals StartRoot r plus 6 EndRoot plus 1 at the given values of the independent variable and simplify.
a.  f left parenthesis negative 6 right parenthesis      b.  f left parenthesis 75 right parenthesis      c.  f left parenthesis x minus 6 right parenthesis
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Part 1
a.  f left parenthesis negative 6 right parenthesisequals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To evaluate the function \( f(r) = \sqrt{r + 6} + 1 \), we will compute the value at the specified inputs.

### Part a: \( f(-6) \)

1. Substitute \(-6\) into the function:
\[
f(-6) = \sqrt{-6 + 6} + 1
\]
2. Simplify inside the square root:
\[
f(-6) = \sqrt{0} + 1
\]
3. Since \(\sqrt{0} = 0\):
\[
f(-6) = 0 + 1 = 1
\]

So,

\[
\boxed{1}
\]

### Part b: \( f(75) \)

1. Substitute \(75\) into the function:
\[
f(75) = \sqrt{75 + 6} + 1
\]
2. Simplify inside the square root:
\[
f(75) = \sqrt{81} + 1
\]
3. Since \(\sqrt{81} = 9\):
\[
f(75) = 9 + 1 = 10
\]

So,

\[
\boxed{10}
\]

### Part c: \( f(x - 6) \)

1. Substitute \(x - 6\) into the function:
\[
f(x - 6) = \sqrt{(x - 6) + 6} + 1
\]
2. Simplify inside the square root:
\[
f(x - 6) = \sqrt{x} + 1
\]

So,

\[
\boxed{\sqrt{x} + 1}
\]

This completes the evaluations for all parts.