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Chapter 1: Problem 11

11-22. For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. [Hint: Use a graphing calculator.] $$ f(x)=\sqrt{x-1} ; \text { find } f(10) $$

Short Answer

Expert verified
f(10) = 3; Domain: [1, 鈭�); Range: [0, 鈭�).

Step by step solution

01

Evaluate the Given Expression

Substitute \( x = 10 \) into the function \( f(x) = \sqrt{x-1} \). This gives \( f(10) = \sqrt{10-1} = \sqrt{9} = 3 \). So, \( f(10) = 3 \).
02

Determine the Domain of the Function

The function is defined where the expression under the square root is non-negative, i.e., \( x-1 \geq 0 \). Solving this inequality, we have \( x \geq 1 \). Thus, the domain of the function is \( [1, \infty) \).
03

Determine the Range of the Function

The function \( f(x) = \sqrt{x-1} \) outputs values based on the square root, which produces non-negative results. As \( x \) starts from 1 and increases, \( \sqrt{x-1} \) starts from 0 and increases without bound. Therefore, the range is \( [0, \infty) \).

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Key Concepts

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

Function Evaluation
Function evaluation is a core concept of calculus and mathematics that centers around finding the output of a function for a specific input. In simpler terms, it's like finding the answer to a question when you're given a specific value to substitute into a function.
Let's take a classical approach to function evaluation using the example from our exercise:
  • A function is given as \( f(x) = \sqrt{x-1} \).
  • To evaluate the function at a specific point, such as \( x = 10 \), we substitute 10 for \( x \) in the expression.
  • This yields \( f(10) = \sqrt{10-1} = \sqrt{9} \).
  • Since the square root of 9 is 3, it follows that \( f(10) = 3 \).

Remember that function evaluation is essential for understanding the behavior and outcome of mathematical models. It helps us determine how different values of \( x \) affect the function as a whole.
Domain of a Function
The domain of a function is a set of all possible input values (usually represented as \( x \)) that the function can accept without resulting in any undefined or non-real outputs. It's like understanding where you can "start" when using a function, such as \( f(x) = \sqrt{x-1} \).
Determining the domain involves ensuring that the function is defined for every value in the domain set:
  • For \( f(x) = \sqrt{x-1} \), the expression inside the square root must be non-negative since square roots of negative numbers are not real.
  • We set up the inequality: \( x-1 \geq 0 \).
  • Solving \( x \geq 1 \), shows that \( x \) can start from 1 and include any number greater.

Therefore, the domain of this function is \([1, \infty)\). This means any number equal to or greater than 1 can be plugged into the function.
Range of a Function
The range of a function is the set of all possible output values (usually represented as \( y \)) that the function can yield. This concept shows us the "reach" of a function on the vertical axis or the top to bottom stretch of its graph.
To find the range of a function like \( f(x) = \sqrt{x-1} \), we consider:
  • The function outputs the square root of a non-negative number, resulting in non-negative outputs as well.
  • For the smallest \( x \) in the domain, i.e., \( x = 1 \), the output is \( f(1) = \sqrt{1-1} = 0 \).
  • As \( x \) increases beyond 1, \( \sqrt{x-1} \) generates outputs starting from 0 and upward indefinitely.

Thus, the range is \([0, \infty)\), indicating that the function can produce any value from 0 and upwards indefinitely. Understanding the range helps us visualize what the output will be for any given input.

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Most popular questions from this chapter

Smoking and Income Based on a recent study, the probability that someone is a smoker decreases with the person's income. If someone's family income is \(x\) thousand dollars, then the probability (expressed as a percentage) that the person smokes is approximately \(y=-0.31 x+40\) (for \(10 \leq x \leq 100)\) a. Graph this line on the window \([0,100]\) by \([0,50]\). b. What is the probability that a person with a family income of $$\$ 40,000$$ is a smoker? [Hint: Since \(x\) is in thousands of dollars, what \(x\) -value corresponds to $$\$ 40,000 ?]$$ c. What is the probability that a person with a family income of $$\$ 70,000$$ is a smoker? Round your answers to the nearest percent.

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$$ \text { If } f(x)=a x, \text { then } f(f(x))=? $$

SOCIAL SCIENCE: Age at First Marriage Americans are marrying later and later. Based on data for the years 2000 to 2007 , the median age at first marriage for men is \(y_{1}=0.12 x+26.8\), and for women it is \(y_{2}=0.12 x+25\), where \(x\) is the number of years since 2000 . a. Graph these lines on the window \([0,30]\) by \([0,35] .\) b. Use these lines to predict the median marriage ages for men and women in the year 2020 . [Hint: Which \(x\) -value corresponds to 2020 ? Then use TRACE, EVALUATE, or TABLE.] c. Predict the median marriage ages for men and women in the year 2030 .

For the quadratic function \(f(x)=a x^{2}+b x+c\), what condition on one of the coefficients will guarantee that the function has a highest value? A lowest value?
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