Chapter 3.1, Problem 61E

Solution

To decide: which of the following functions is exponential.

E

Given information:

The functions:

  A. $f\left(x\right)=a^2$

  B. $f\left(x\right)=x^3$

  C. $f\left(x\right)=x^{\frac{2}{3}}$

  D. $f\left(x\right)=\sqrt[3]{x}$

  E. $f\left(x\right)=8^x$

Definition Used:

The general form of an exponential function f ( x ) is

  $f\left(x\right)=a\cdot b^x$   ..... (1)

Where $a\ne 0$, b is a positive real number called the base of the function and $b\ne 1$ .

Here, x is a variable and is called the exponent.

Explanation:

Consider the functions,

a. $f\left(x\right)=a^2$

Since, a is a constant, so this gives that $a^2$ is also constant, which implies that $f\left(x\right)=a^2$ is a constant function.

b. $f\left(x\right)=x^3$

This is a function of the form $f\left(x\right)=x^n$, so it is a power function.

c. $f\left(x\right)=x^{\frac{2}{3}}$

This is a function of the form $f\left(x\right)=x^n$, so it is also a power function.

d. $f\left(x\right)=\sqrt[3]{x}$

This can be rewritten as:

  $f\left(x\right)=\sqrt[3]{x}=x^{\frac{1}{3}}$

This is a function of the form $f\left(x\right)=x^n$, so it is also a power function.

e. $f\left(x\right)=8^x$

This function satisfies the definition of the exponential function given above and it has the base $b=8$ .

So, it is an exponential function.

Out of all given options, only option E is an exponential function.