Problem 33 - Contemporary Precalculus

Question

Find the difference quotient of the function. $$f(x)=10^{x}$$

Step-by-Step Solution

Answer

Answer: The difference quotient of the function f(x) = 10^x is $$\frac{10^x(10^h - 1)}{h}$$.

Step 1: Determine f(x + h)

The given function is: $$f(x) = 10^x$$. To find f(x + h), replace x with (x + h) in the function: $$f(x + h) = 10^{x + h}$$ Step 2: Plug f(x + h) and f(x) into the Difference Quotient Formula

Step 2: Apply the difference quotient formula

Using the difference quotient formula, we plug in our values for f(x + h) and f(x). The formula is: $$\frac{f(x + h) - f(x)}{h}$$ and now we insert f(x + h) and f(x): $$\frac{10^{x + h} - 10^x}{h}$$ Step 3: Simplify the Difference Quotient

Step 3: Simplify the expression

Using the properties of exponents, we can rewrite the expression: $$\frac{10^x\cdot10^h - 10^x}{h}$$ Now, factor out 10^x from the numerator to simplify further: $$\frac{10^x(10^h - 1)}{h}$$ This is the simplified difference quotient of the given function: $$\frac{10^x(10^h - 1)}{h}$$

Key Concepts

Exponential Functions

An exponential function is one in which the variable appears in the exponent. They have the general form:

• \( f(x) = a^{x} \)

where \( a \) is a positive constant, not equal to 1.
These functions show growth or decay behavior very quickly and are commonly used to model real-world situations like population growth or radioactive decay.
For example, our function here, \( f(x) = 10^x \), is an exponential function where the base of the exponent is 10.
The base, 10, indicates that the function grows by a factor of 10 for every unit increase in \( x \).
This leads to rapid growth as \( x \) increases.
Understanding exponential functions is crucial for finding the difference quotient in this exercise because they directly define how \( f(x) \) changes as \( x \) changes.

Simplifying Expressions

Simplifying expressions involves rewriting them in their simplest form. This concept is crucial to solving problems efficiently and accurately.
Simplifying can include combining like terms, factoring, or using algebraic identities.
When dealing with the difference quotient for an exponential function like \( f(x) = 10^x \), simplification focuses on breaking down the terms properly:

• Write \( f(x + h) = 10^{x + h} \) as \( 10^x \cdot 10^h \)
• Insert this into the difference quotient formula: \( \frac{10^x \cdot 10^h - 10^x}{h} \)
• Factor out \( 10^x \) from the numerator to simplify to \( \frac{10^x(10^h - 1)}{h} \)

This process of simplification makes it much easier to interpret and work with the expression, especially when analyzing the behavior of the function.

Properties of Exponents

The properties of exponents are vital for handling expressions that involve exponential functions. Here, we use them to simplify and manipulate exponentials:

• Product of Powers: When multiplying like bases, add the exponents:\[ a^m \cdot a^n = a^{m+n} \]
• Quotient of Powers: When dividing like bases, subtract the exponents:\[ \frac{a^m}{a^n} = a^{m-n} \]
• Power of a Power: Apply the exponent outside the parentheses to each term inside:\[ (a^m)^n = a^{m \cdot n} \]

In the context of our exercise, we directly apply the product of powers property. We rewrote \( 10^{x+h} = 10^x \cdot 10^h \) during simplification.
Also, recognizing that \( 10^x \) can be factored out effectively used these properties to simplify our expression further:\( \frac{10^x(10^h - 1)}{h} \).
These properties make working with exponential functions easier, ensuring calculations remain accurate and manageable.