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Chapter 4: Problem 20

Evaluate the functions at the given values of \(x .\) Round to 4 decimal places if necessary. \(g(x)=7^{x}\) a. \(g(-2)\) b. \(g(5.9)\) c. \(g(\sqrt{11})\) d. \(g(e)\)

Short Answer

Expert verified
g(-2) = 0.0204, g(5.9) = 341422.8530, g(\sqrt{11}) = 10194.1054, g(e) = 678.0266

Step by step solution

01

Understand the Function

Given the function is \(g(x) = 7^x\). This means for any value of \(x\), you need to raise 7 to the power of \(x\).
02

Calculate \(g(-2)\)

Substitute \(-2\) for \(x\) in the function: \(g(-2) = 7^{-2} = \frac{1}{7^2} = \frac{1}{49} \approx 0.0204\).
03

Calculate \(g(5.9)\)

Substitute \(5.9\) for \(x\) in the function: \(g(5.9) = 7^{5.9} \approx 341422.8530\).
04

Calculate \(g(\sqrt{11})\)

Substitute \(\sqrt{11}\) for \(x\) in the function: \(g(\sqrt{11}) = 7^{\sqrt{11}} \approx 10194.1054\).
05

Calculate \(g(e)\)

Substitute \(e\) (where \(e \approx 2.7183\)) for \(x\) in the function: \(g(e) = 7^e \approx 678.0266\).

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Key Concepts

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

Function Evaluation
Evaluation of a function involves substituting the given values of the independent variable, typically denoted as \(x\), into the function's formula and calculating the result. For the function \(g(x) = 7^x\):
  • To evaluate \(g(-2)\), we substitute \(-2\) for \(x\) to get \(7^{-2}\). This results in calculating the reciprocal of \(7^2\) or \(49\), so \(g(-2) = \frac{1}{49} \approx 0.0204\).
  • To evaluate \(g(5.9)\), we substitute \(5.9\) for \(x\), leading to \(7^{5.9} \approx 341422.8530\).
  • With \(g(\text{sqrt}(11))\), we use the square root of \(11\) for \(x\), so \(g(\text{sqrt}(11)) = 7^{{\sqrt{11}}} \approx 10194.1054\).
  • Finally, for \(g(e)\), where \(e \approx 2.7183\), we get \(g(e) = 7^e \approx 678.0266\).
Function evaluation is fundamental to understanding how functions respond to different inputs.
Exponents
Exponents or powers refer to the number of times a number, called the base, is multiplied by itself. In the function \(g(x) = 7^x\), \(7\) is the base and \(x\) is the exponent. Some key ideas about exponents include:
  • An exponent of 2 means squaring the base: \(7^2 = 49\).
  • An exponent of -2, like in \(g(-2)\), means taking the reciprocal: \(7^{-2} = \frac{1}{49} = 0.0204\).
  • Fractional exponents involve roots. For instance, \(7^{0.5} = \sqrt{7}\).
  • Exponential growth can be represented when the exponent is positive, such as \(7^{5.9}\) which results in a very large number \(341422.8530\).
Exponents are critical in representing repeated multiplication and complex growth behavior.
Irrational Numbers
Irrational numbers are real numbers that cannot be written as a simple fraction. They have non-terminating, non-repeating decimal representations. Examples include:
  • The square root of non-perfect squares, like \(\sqrt{11}\). This is used in the calculation of \(g(\sqrt{11}) = 7^{\sqrt{11}} \approx 10194.1054\).
  • \(e\), known as Euler's number, which is approximately \(2.7183\). It's used in the mathematical constant e\ for \(g(e) = 7^e \approx 678.0266\).
Understanding irrational numbers is crucial as they frequently appear in functions and various mathematical contexts. They reveal the complexity and beauty of numbers that can't be precisely represented by fractions.
Euler's Number
Euler's number, denoted as \(e\), is an irrational number approximately equal to \(2.7183\). It's a fundamental constant in mathematics, particularly in calculus and complex analysis. Here are some important properties of \(e\):
  • It's the base of natural logarithms.
  • Appears in compound interest calculations, described by the formula \(A = P(1 + \frac{r}{n})^{nt}\) which approaches \(Pe^{rt}\).
  • Functions of the form \(e^x\) arise in various growth and decay problems, such as population growth and radioactive decay.
For our function evaluation, \(g(e) = 7^e \approx 678.0266\). This demonstrates how \(e\) is used as an exponent and highlights its relevance in exponential functions.

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Most popular questions from this chapter

Technetium- \(99\left({ }^{99 \mathrm{~m}} \mathrm{Tc}\right)\) is a radionuclide used widely in nuclear medicine. \({ }^{99 \mathrm{~m}} \mathrm{Tc}\) is combined with another substance that is readily absorbed by a targeted body organ. Then, special cameras sensitive to the gamma rays emitted by the technetium are used to record pictures of the organ. Suppose that a technician prepares a sample of \(^{99 \mathrm{~m}}\) Tc-pyrophosphate to image the heart of a patient suspected of having had a mild heart attack. a. At noon, the patient is given \(10 \mathrm{mCi}\) (millicuries) of \({ }^{99 \mathrm{~m}} \mathrm{Tc}\). If the half-life of \({ }^{99 \mathrm{~m}} \mathrm{Tc}\) is \(6 \mathrm{hr}\), write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the radioactivity level \(Q(t)\) after \(t\) hours. b. At what time will the level of radioactivity reach \(3 \mathrm{mCi} ?\) Round to 1 decimal place.

Compare the graphs of the functions. $$ \mathrm{Y}_{1}=\ln \left(\frac{x}{2}\right) \quad \text { and } \quad \mathrm{Y}_{2}=\ln x-\ln 2 $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(2^{1-6 x}=7^{3 x+4}\)

Use the model \(A=P e^{r t} .\) The variable \(A\) represents the future value of \(P\) dollars invested at an interest rate \(r\) compounded continuously for \(t\) years. If \(\$ 10,000\) is invested in an account earning \(5.5 \%\) interest compounded continuously, determine how long it will take the money to triple. Round to the nearest year.

9\. The population of the United States \(P(t)\) (in millions) since January 1,1900 , can be approximated by $$P(t)=\frac{725}{1+8.295 e^{-0.0165 t}}$$ where \(t\) is the number of years since January \(1,1900 .\) (See Example 6\()\) a. Evaluate \(P(0)\) and interpret its meaning in the context of this problem. b. Use the function to predict the U.S. population on January \(1,2020 .\) Round to the nearest million. c. Use the function to predict the U.S. population on January 1,2050 . d. Determine the year during which the U.S. population will reach 500 million. e. What value will the term \(\frac{8.295}{e^{0.0165 t}}\) approach as \(t \rightarrow \infty\) ? f. Determine the limiting value of \(P(t)\).
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