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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 33c
Chapter 3, Problem 33c

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

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1
Identify the given function: \(f(r) = \sqrt{r + 6} + 3\).
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.
Substitute \(r\) with \(x - 6\) in the function: \(f(x - 6) = \sqrt{(x - 6) + 6} + 3\).
Simplify the expression inside the square root: \((x - 6) + 6 = x\).
Write the simplified function: \(f(x - 6) = \sqrt{x} + 3\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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).
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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.
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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.
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Related Practice
Textbook Question

Find f/g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Textbook Question

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(r) = √(r + 6) +3 b. f(10)

Textbook Question

Find fg and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Textbook Question

Use the graph of y = f(x) to graph each function g. g(x) = f(x+2)

Textbook Question

Find ƒ+g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Textbook Question

In Exercises 33–34, find and simplify the difference quotient [f(x + h) - f(x)]/h, h =/= 0 for the given function. f(x) = -2x^2 + x + 10