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Chapter 4: Problem 82

Explain why the \(f(x)=x^{2}\) is not an exponential function.

Short Answer

Expert verified
The variable is in the base for \(f(x)=x^{2}\), not in the exponent.

Step by step solution

01

Understanding Exponential Functions

Exponential functions have the form \(f(x)=a \cdot b^{x}\) where \(a\) is a constant and \(b\) is the base of the exponential, which must be positive and not equal to 1.
02

Identify the Given Function

The given function is \(f(x) = x^{2}\).
03

Compare Formats

Compare the format of the given function \(f(x)=x^{2}\) with the general form of an exponential function \(f(x)=a \cdot b^{x}\). Notice that in \(f(x)=x^{2}\), the variable \(x\) is in the base, not in the exponent.
04

Conclude the Difference

Since \(f(x)=x^{2}\) does not fit the form \(a \cdot b^{x}\), it cannot be considered an exponential function.

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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 special type of mathematical function that grow very rapidly. They have the general form: \( f(x) = a \cdot b^{x} \) where:
  • a is a constant.
  • b is the base of the exponential function, which must be a number greater than 0 and not equal to 1.
  • x is the exponent, which is typically a variable.

The key characteristic of exponential functions is that the variable is in the exponent. This means as the value of x increases, the function鈥檚 value grows at an increasingly faster rate. One common example of an exponential function is the compound interest formula in finance.
polynomial functions
Polynomial functions are another type of function, widely used in mathematics. They are expressed in the form of: \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \) where:
  • Each a_i is a coefficient, which can be any real number.
  • The highest power of x (i.e., n) determines the degree of the polynomial.
  • The degree of the polynomial defines its highest growth rate as x increases.

For example, a quadratic function \( f(x) = x^2 \) is a polynomial of degree 2. It鈥檚 crucial to note that in polynomial functions, the variable is the base raised to different powers, rather than being the exponent like in exponential functions.
function comparison
Understanding the differences between exponential and polynomial functions is essential. Let's delve into these differences:
  • Growth Rate: Exponential functions grow much faster than polynomial functions as x increases. For instance, compare \( 2^x \) (exponential) with \( x^2 \) (polynomial). As x becomes very large, \( 2^x \) will surpass \( x^2 \) rapidly.
  • Variable Position: In exponential functions, the variable is in the exponent (e.g., \( 3^x \)). In polynomial functions, the variable is in the base being raised to a power (e.g., \( x^3 \)).
  • Expression Forms: Typical forms of exponential functions are \( f(x) = a \cdot b^x \), while polynomial functions follow forms like \( f(x) = x^n + \text{lower degree terms} \).

Using the exercise's example, the function \( x^2 \) is clearly a polynomial because the variable x is the base raised to the power of 2. It does not match the format of an exponential function, where x would need to be the exponent for rapid growth.

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Most popular questions from this chapter

Beginning on January 1 , park rangers in Everglades National Park began recording the water level for one particularly dry area of the park. The water level was initially \(2.5 \mathrm{ft}\) and decreased by approximately \(0.015 \mathrm{ft} /\) day. a. Write a function representing the water level \(L(x)\) (in \(\mathrm{ft}\) ), \(x\) days after January \(1 .\) b. Write an equation for \(L^{-1}(x)\). c. What does the inverse function represent in the context of this problem? d. Evaluate \(L^{-1}(1.9)\) and interpret its meaning in context.

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 ). $$ \log _{5}\left(\frac{1}{x}\right)=\frac{1}{\log _{5} x} $$

Explain why the domain of \(f(x)=x^{2}+k\) must be restricted to find an inverse function.

Write \(\ln (x+4)=6\) in exponential form.

a. The populations of two countries are given for January 1,2000 , and for January 1,2010 . Write a function of the form \(P(t)=P_{0} e^{k t}\) to model each population \(P(t)\) (in millions) \(t\) years after January 1, 2000.$$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Population } \\ \text { in 2000 } \\ \text { (millions) } \end{array} & \begin{array}{c} \text { Population } \\ \text { in 2010 } \\ \text { (millions) } \end{array} & \boldsymbol{P}(t)=\boldsymbol{P}_{0} e^{k t} \\ \hline \text { Switzerland } & 7.3 & 7.8 & \\ \hline \text { Israel } & 6.7 & 7.7 & \\ \hline \end{array}$$ b. Use the models from part (a) to predict the population on January \(1,2020,\) for each country. Round to the nearest hundred thousand. c. Israel had fewer people than Switzerland in the year 2000 , yet from the result of part (b), Israel will have more people in the year \(2020 ?\) Why? d. Use the models from part (a) to predict the year during which each population will reach 10 million if this trend continues.
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