/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Evaluate the functions at the gi... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 21

Evaluate the functions at the given values of \(x .\) Round to 4 decimal places if necessary. \(h(x)=\left(\frac{1}{4}\right)^{x}\) a. \(h(-3)\) b. \(h(1.4)\) c. \(h(\sqrt{3})\) d. \(h(0.5 e)\)

Short Answer

Expert verified
a. 64 \ b. 0.1056 \ c. 0.1513 \ d. 0.0996

Step by step solution

01

Substitute for part (a)

Evaluate the function at \(x = -3\). Substitute \(x = -3\) into \(h(x) = \left(\frac{1}{4}\right)^{x}\)\: \ h(-3) = \left(\frac{1}{4}\right)^{-3} \
02

Simplify for part (a)

To simplify \(\left(\frac{1}{4}\right)^{-3} \), use the rule \(a^{-n} = \frac{1}{a^{n}}\): \ \left(\frac{1}{4}\right)^{-3} = 4^{3} = 64 \
03

Substitute for part (b)

Evaluate the function at \(x = 1.4\). Substitute \(x = 1.4\) into \(h(x) = \left(\frac{1}{4}\right)^{x}\): \h(1.4) = \left(\frac{1}{4}\right)^{1.4} \
04

Use a calculator for part (b)

Using a calculator, compute \(\left(\frac{1}{4} \right)^{1.4}\). The result is approximately: \ h(1.4) \approx 0.1056 \ (rounded to 4 decimal places)
05

Substitute for part (c)

Evaluate the function at \(x = \sqrt{3}\). Substitute \(x = \sqrt{3}\) into \(h(x) = \left(\frac{1}{4}\right)^{x}\): \ h(\sqrt{3}) = \left(\frac{1}{4}\right)^{\sqrt{3}} \
06

Use a calculator for part (c)

Using a calculator, compute \(\left(\frac{1}{4} \right)^{\sqrt{3}}\). The result is approximately: \h(\sqrt{3}) \approx 0.1513 \ (rounded to 4 decimal places) \
07

Substitute for part (d)

Evaluate the function at \(x = 0.5e\). Substitute \(x = 0.5e\) into \(h(x) = \left(\frac{1}{4}\right)^{x}\): \h(0.5e) = \left(\frac{1}{4}\right)^{0.5e} \
08

Use a calculator for part (d)

Using a calculator, compute \(\left(\frac{1}{4} \right)^{0.5e}\). The result is approximately: \h(0.5e) \approx 0.0996 \ (rounded to 4 decimal places)

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Key Concepts

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

Exponential Functions
Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. They are written in the form: \(f(x) = a^x\).
In this context, our function is \(h(x) = \left( \frac{1}{4} \right)^x\), where the base is \(\frac{1}{4}\) and the exponent is \(x\).
Exponential functions grow or decay rapidly based on the value of the exponent, making them very different from linear functions.
These functions are common in various fields such as finance, biology, and physics.
Function Substitution
Function substitution involves replacing the variable in a function with a specific value. In this exercise, we are asked to evaluate the function \(h(x) = \left( \frac{1}{4} \right)^x\) at different values of \(x\).
For example, in part (a), we substitute \(x = -3\) into the function to get \(h(-3) = \left( \frac{1}{4} \right)^{-3}\).
Similarly, for parts (b), (c), and (d), we substitute \(x = 1.4\), \(x = \sqrt{3}\), and \(x = 0.5e\) respectively.
Substitution is essential as it allows us to evaluate the function at specific points.
Rounding Numbers
Rounding numbers is an important skill in mathematics, especially when working with approximations. In this exercise, we round our results to four decimal places.
For instance, after evaluating \(h(1.4)\) using a calculator, we get approximately \(0.1056\). This means that we round 0.105600000 to 0.1056.
Rounding ensures that results are manageable and easy to interpret, and it is often done to maintain a standard level of accuracy.
Always remember to follow the rounding rules: if the fifth decimal place is 5 or more, round up the fourth place by one, otherwise, leave it as it is.
Calculator Usage
Using a calculator is essential for evaluating complex expressions, especially those involving non-integer exponents. To evaluate \(\left(\frac{1}{4}\right)^x\), we rely on a calculator for accuracy.
Here are a few tips for using a calculator effectively:
  • Confirm that your calculator is in the correct mode (usually 'normal' or 'scientific').
  • Use parentheses to ensure the correct order of operations.
  • Double-check your input to avoid errors.
For example, to calculate \(\left(\frac{1}{4}\right)^{\sqrt{3}}\), enter \((1/4)^{\sqrt{3}}\) on your calculator and get an approximate value of 0.1513.
Practice using your calculator with various expressions to become more familiar with its functions.

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Most popular questions from this chapter

Suppose that \(\$ 50,000\) from a retirement account is invested in a large cap stock fund. After 20 yr, the value is \(\$ 194,809.67\). a. Use the model \(A=P e^{r t}\) to determine the average rate of return under continuous compounding. b. How long will it take the investment to reach onequarter million dollars? Round to 1 decimal place.

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{2}(7 y)+\log _{2} 1=\log _{2}(7 y) $$

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{8} 5 $$

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{2} 15 $$

A function of the form \(P(t)=a b^{t}\) represents the population of the given country \(t\) years after January 1,2000 . a. Write an equivalent function using base \(e\); that is, write a function of the form \(P(t)=P_{0} e^{k t} .\) Also, determine the population of each country for the year 2000 . $$\begin{array}{|l|c|c|c|} \hline \text { Country } & P(t)=a b^{t} & P(t)=P_{0} e^{k t} & \begin{array}{c} \text { Population } \\ \text { in } 2000 \end{array} \\ \hline \text { Haiti } & P(t)=8.5(1.0158)^{t} & & \\ \hline \text { Sweden } & P(t)=9.0(1.0048)^{t} & & \\ \hline \end{array}$$ b. The population of the two given countries is very close for the year 2000 , but their growth rates are different. Determine the year during which the population of each country will reach 10.5 million. c. Haiti had fewer people in the year 2000 than Sweden. Why did Haiti reach a population of 10.5 million sooner?
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