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Chapter 0: Problem 30

\(f(x)=\sqrt{x+8}+2\) (a) \(f(-8)\) (b) \(f(1)\) (c) \(f(x-8)\)

Short Answer

Expert verified
\(f(-8) = 2\), \(f(1) = 5\), and \(f(x-8) = \sqrt{x} + 2\)

Step by step solution

01

Evaluate \(f(-8)\)

The expression under the square root should be non-negative, so it's safe to substitute \(-8\) for \(x\). So, \(f(-8) = \sqrt{-8+8} + 2 = \sqrt{0} + 2 = 0 + 2 = 2\).
02

Evaluate \(f(1)\)

Substitute \(x = 1\) in \(f(x)\). So, \(f(1) = \sqrt{1+8}+2 = \sqrt{9} + 2 = 3 + 2 = 5\).
03

Evaluate \(f(x-8)\)

Substitute \(x = x - 8\) in \(f(x)\). So, \(f(x-8) = \sqrt{x-8+8}+2 = \sqrt{x} + 2\). Note that here, the function will only be defined where \(x - 8 \geq 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
A square root function is an expression involving the square root of a variable. In the function \(f(x) = \sqrt{x+8} + 2\), the square root component is \(\sqrt{x+8}\). Square root functions are important in mathematics because they are used to determine a value that, when multiplied by itself, yields the original number.
  • The expression inside the square root, in this case \(x+8\), is called the radicand.
  • Square roots can only be performed on non-negative numbers within the realm of real numbers, making the expression inside always greater than or equal to zero.
In practical terms, square root functions often result in a graph that forms a gentle upward curve. When evaluating square root functions, calculating the actual operation is crucial in deriving the value of \(f(x)\), which in our example, also includes adding 2 to the square root calculation.
Domain of a Function
The domain of a function refers to all the possible input values (typically \(x\) values) that are acceptable and make the function work without any mathematical errors. For the function \(f(x) = \sqrt{x+8} + 2\), it's essential for the radicand \(x+8\) to be greater than or equal to zero so that \(\sqrt{x+8}\) is real.
  • We set the inequality \(x+8 \geq 0\).
  • Simplifying, we find that \(x \geq -8\).
This means the domain of the function \(f(x)\) is all real numbers \(x\) such that \(x \geq -8\). Restricting the domain ensures that we only consider inputs where the function produces valid and real-number outputs. Always take care when performing operations on functions such as substitution, which can affect the function's overall domain.
Function Substitution
Function substitution involves replacing the variable in a function with another expression or specific number to evaluate or transform the function. In our examples, we see substitutions like \(f(-8)\), \(f(1)\), and \(f(x-8)\).
  • For \(f(-8)\), substitute \(x = -8\) into \(f(x)\) to get \(f(-8) = \sqrt{-8+8} + 2 = 2\).
  • For \(f(1)\), substitute \(x = 1\) yielding \(f(1) = \sqrt{1+8} + 2 = 5\).
  • In \(f(x-8)\), substitute \(x = x-8\): \(f(x-8) = \sqrt{x-8+8} + 2 = \sqrt{x} + 2\).
Function substitution is a powerful tool as it lets you explore how different inputs or transformations affect the function. When substituting, always pay attention to how it affects the domain of the function, especially if the substitution introduces new terms that require additional consideration.

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Most popular questions from this chapter

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ h(x)=-\frac{4}{x^{2}} $$

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. If the inverse function of \(f\) exists and the graph of \(f\) has a \(y\)-intercept, the \(y\)-intercept of \(f\) is an \(x\)-intercept of \(f^{-1}\).

In Exercises 75-78, use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ f^{-1} \circ g^{-1} $$

Average Price The average prices \(p\) (in thousands of dollars) of a new mobile home in the United States from 1990 to 2002 (see figure) can be approximated by the model $$ p(t)= \begin{cases}0.182 t^{2}+0.57 t+27.3, & 0 \leq t \leq 7 \\ 2.50 t+21.3, & 8 \leq t \leq 12\end{cases} $$ where \(t\) represents the year, with \(t=0\) corresponding to 1990. Use this model to find the average price of a mobile home in each year from 1990 to 2002 . (Source: U.S. Census Bureau)

Transportation For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate \(=8-0.05(n-80), \quad n \geq 80\) where the rate is given in dollars and \(n\) is the number of people. (a) Write the revenue \(R\) for the bus company as a function of \(n\). (b) Use the function in part (a) to complete the table. What can you conclude? \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline\(n\) & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\ \hline\(R(n)\) & & & & & & & \\ \hline \end{tabular}
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