Problem 1 Which functions are exponential?... [FREE SOLUTION]

Chapter 4: Problem 1

Which functions are exponential? a. \(f(x)=\left(\frac{1}{\sqrt{3}}\right)^{x}\) b. \(f(x)=1^{x}\) c. \(f(x)=x^{\sqrt{3}}\) d. \(f(x)=(-2)^{x}\) e. \(f(x)=\pi^{x}\)

Short Answer

Expert verified

The exponential functions are: (a) \(f(x) = \left(\frac{1}{\sqrt{3}}\right)^{x}\) and (e) \(f(x) = \pi ^ {x}\).

Step by step solution

- Understand Exponential Functions

An exponential function can be written in the form \(f(x) = a^x\), where \(a\) is a positive constant base, and \(x\) is the exponent. The base \(a\) must be a positive real number and different from 1.

- Analyze each function

Evaluate each given function based on the definition of an exponential function.

- Function a

The function \(f(x)=\left(\frac{1}{\sqrt{3}}\right)^{x}\) is in the form of \(a^x\) where the base \(a=\frac{1}{\sqrt{3}}\) is positive and not equal to 1. Thus, this is an exponential function.

- Function b

The function \(f(x)=1^x\) has a base of 1. Since exponential functions require the base to be different from 1, this is not an exponential function.

- Function c

The function \(f(x)=x^{\sqrt{3}}\) has a variable base and a constant exponent. For a function to be exponential, the base must be constant and the exponent must be the variable. Therefore, this is not an exponential function.

- Function d

The function \(f(x)=(-2)^x\) has a negative base. Exponential functions require the base to be a positive real number; hence this is not an exponential function.

- Function e

The function \(f(x)=\pi^x\) is in the form \(a^x\) where the base \(a=\pi\) is a positive constant and different from 1. Therefore, this is an exponential function.

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Key Concepts

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

Exponential Function Definition

An exponential function is a mathematical expression where the variable is in the exponent. Typically, it's written in the form: \(f(x) = a^x\). Here, \(a\) represents a positive constant base while \(x\) is the variable exponent. This structure is key to identifying exponential functions.

For a function to qualify as exponential:

The base \(a\) must be a positive real number.
The base \(a\) must be different from 1.
The exponent \(x\) must be the variable.

If the function meets these criteria, it is classified as an exponential function. This distinction helps distinguish these functions from other types of mathematical functions.

Variable Exponent

The concept of having a variable exponent is crucial in defining exponential functions. In an exponential function, the variable \(x\) appears in the exponent position. This can be observed in functions of the form \(f(x) = a^x\), where \(x\) is the changing element affecting the growth rate.

For example, in the function \(f(x) = 2^x\):

When \(x = 1\), the function value is 2.
When \(x = 2\), the function value is 4.
When \(x = 3\), the function value is 8.

Notice how the value of the function changes as \(x\) changes? This dynamic growth is a characteristic feature of exponential functions and is driven by the variable exponent.

Positive Constant Base

In exponential functions, the base \(a\) plays a critical role in determining the nature of the function. To be an exponential function, the base must be:

A positive real number.
Different from 1.

These conditions ensure the function grows at a consistent exponential rate. For instance:

In \(f(x)=3^x\), the base is 3, a positive number different from 1, so this is an exponential function.
In \(f(x)=1^x\), the base is 1, making it constant and not truly exponential.
In \(f(x)=(-2)^x\), the base is negative, disqualifying it as an exponential function.

By ensuring the base is a positive constant (and not 1), exponential functions maintain their distinct and predictable patterns of growth.

Non-Exponential Functions

Not all functions are exponential, even if they visually seem to grow or shrink significantly. To clarify some misconceptions:

\(f(x) = x^{\frac{3}{2}}\) is not an exponential function because the base is the variable, not a constant.
\(f(x) = (-2)^x\) is not an exponential function because the base is negative.
\(f(x) = 1^x\) is not an exponential function as the base is 1.

Recognizing non-exponential functions helps avoid confusion. The critical requirement remains: the base must be a positive, non-1 constant, and the exponent must be the variable. These distinctions aid in categorizing and understanding different types of mathematical functions accurately.

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91影视

Which functions are exponential? a. \(f(x)=\left(\frac{1}{\sqrt{3}}\right)^{x}\) b. \(f(x)=1^{x}\) c. \(f(x)=x^{\sqrt{3}}\) d. \(f(x)=(-2)^{x}\) e. \(f(x)=\pi^{x}\)

Short Answer

Step by step solution

- Understand Exponential Functions

- Analyze each function

- Function a

- Function b

- Function c

- Function d

- Function e

Key Concepts

Exponential Function Definition

Variable Exponent

Positive Constant Base

Non-Exponential Functions

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