Problem 40 - Algebra and Trigonometry

Question

If \(f(x)=10^{x},\) show that \(\frac{f(x+h)-f(x)}{h}=10^{x}\left(\frac{10^{h}-1}{h}\right)\)

Step-by-Step Solution

Answer

The given expression simplifies to the desired form by factoring out \(10^x\).

Step 1: Substitute the function into the expression

We start by substituting the given function \(f(x) = 10^x\) into the expression \(\frac{f(x+h) - f(x)}{h}\). This gives us \(\frac{10^{x+h} - 10^x}{h}\).

Step 2: Simplify the expression

The expression \(\frac{10^{x+h} - 10^x}{h}\) can be further simplified by factoring out \(10^x\). Using the property \(10^{x+h} = 10^x \cdot 10^h\), we have \(\frac{10^x (10^h - 1)}{h}\).

Step 3: Factor and rearrange terms

Now, extract the common factor \(10^x\) from the numerator: \(10^x \cdot \frac{10^h - 1}{h}\). This simplifies the expression to the form \(10^x \left( \frac{10^h - 1}{h} \right)\).

Step 4: Verify the result

After simplification, we confirm that the expression matches the required result: \(\frac{f(x+h) - f(x)}{h} = 10^x \left( \frac{10^h - 1}{h} \right)\), which proves the problem statement.

Key Concepts

Exponential Functions

Exponential functions are a special type of mathematical function characterized by a constant base raised to a variable exponent. In our example, the function is defined as \(f(x) = 10^x\). This means the base is 10, and the power to which 10 is raised is the variable \(x\).
Exponential functions are widely used in various fields including finance, science, and engineering because they model scenarios where rates of change are proportional to value such as compound interest or population growth.

• The base \(b\) is always a positive real number.
• Exponential functions exhibit rapid growth or decay.

The general form is \(f(x) = a \cdot b^x\), where \(a\) is a constant multiplier, and \(b\) is the base. In our function, \(a = 1\) and \(b = 10\). Understanding how these components interact helps in analyzing changes over time.

Function Notation

Function notation provides a simple way to indicate mathematical functions and their outputs in terms of inputs. In the expression \(f(x) = 10^x\), \(f\) is the function name, and \(x\) is the input variable.
The notation \(f(x+h)\) signifies the function applied to an input \(x + h\), which effectively shifts the function horizontally by \(h\). This is crucial in expressions involving differences, such as the Difference Quotient, as seen in the given exercise.
Using function notation, you can easily substitute different values or expressions to explore how outputs change, which is vital in calculus for concepts such as derivatives and limits.

Algebraic Manipulation

Algebraic manipulation involves rearranging expressions to simplify or solve them. It's like transforming pieces of a puzzle to see a clearer picture. For example, the expression \(\frac{f(x+h) - f(x)}{h}\) in the exercise requires manipulation to reach a simpler form.
A key technique here is substituting \(10^x\) to factor the expression \(10^{x+h} - 10^x\) to \(10^x (10^h - 1)\).
At this point, recognizing patterns or common factors, like \(10^x\), allows you to simplify expressions. Using identities and properties of exponents also contributes to efficient algebraic manipulation. The ability to simplify terms systematically is invaluable in advanced mathematics.

Factoring Expressions

Factoring is a method used to break down complex expressions into simpler components, often making solving problems easier or revealing important properties. In this case, \(10^{x+h} - 10^x\) was factored as \(10^x (10^h - 1)\).
This factoring process hinges on recognizing the common base, \(10^x\), across the terms, illustrating the relationship between them. By factoring, we simplify the difference quotient to a form that is easily verifiable.
Factoring can also involve recognizing and applying algebraic identities like the difference of squares or grouping terms strategically. This skill is not only beneficial in simplifying expressions but is also key in solving equations efficiently across algebra and calculus.