Problem 72 - College Algebra with Modeling and Visualization

Question

Complete the following for the given \(f(x)\) (a) Find \(f(x+h)\) (b) Find the difference quotient of \(f\) and simplify. $$ f(x)=10 x $$

Step-by-Step Solution

Answer

a) \(f(x+h) = 10x + 10h\); b) The difference quotient simplifies to 10.

Step 1: Substitute (x+h) into f(x)

To find \(f(x + h)\), substitute \(x + h\) into the function \(f(x) = 10x\). This means we replace every \(x\) with \(x + h\):\[f(x + h) = 10(x + h) = 10x + 10h\].

Step 2: Compute the Difference Quotient

The difference quotient formula is given by:\[\frac{f(x+h) - f(x)}{h}\]Substitute \(f(x+h) = 10x + 10h\) and \(f(x) = 10x\):\[\frac{(10x + 10h) - 10x}{h}\].

Step 3: Simplify the Difference Quotient

Simplify the expression from Step 2:1. Subtract \(10x\) from \(10x + 10h\). This gives us:\[\frac{10h}{h}\]2. Divide \(10h\) by \(h\), simplifying to:\[10\].

Key Concepts

Function Substitution

Function substitution is a fundamental concept in calculus used to evaluate functions at different values. For our function, which is defined as \( f(x) = 10x \), substitution involves replacing every instance of \( x \) with another expression. In this context, you perform function substitution by replacing \( x \) with \( x + h \). This process requires substituting \( (x + h) \) into the function to find \( f(x + h) \).

• Start with the original function: \( f(x) = 10x \)
• Substitute \( x + h \) into the function: \( f(x + h) = 10(x + h) \)
• Distribute the multiplication across the brackets: \( f(x + h) = 10x + 10h \)

This process is crucial for understanding how functions change with shifts and transformations. It sets the stage for more complex operations like differentiation, where tiny changes in \( x \) are analyzed.

Algebraic Simplification

Algebraic simplification is the process of reducing complex mathematical expressions to their simplest form. This technique plays a pivotal role in handling the difference quotient. The difference quotient for the function \( f(x) = 10x \) involves simplifying an expression derived from two function evaluations: \( f(x + h) \) and \( f(x) \).To simplify the difference quotient \( \frac{f(x+h) - f(x)}{h} \), follow these steps:

• Combine the function values: Substitute \( f(x+h) = 10x + 10h \) and \( f(x) = 10x \) into the difference quotient formula.
• The result is: \( \frac{(10x + 10h) - 10x}{h} \).
• Subtract \( 10x \) from \( 10x + 10h \): this removes \( 10x \), leaving \( 10h \).
• Now, simplify the expression to: \( \frac{10h}{h} \).
• Divide by \( h \), yielding simply: \( 10 \).

This process makes the expression manageable and reveals fundamental linear relationships in the function, paving the way for derivative calculations.

Linear Functions

We're dealing with a linear function in the form \( f(x) = 10x \). Linear functions are characterized by having a constant rate of change and are represented graphically by straight lines. The core property of linear functions is that they can always be expressed in the format \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.For the function \( f(x) = 10x \):

• The coefficient \( 10 \) represents the slope \( m \), indicating that for every unit increase in \( x \), \( f(x) \) increases by 10.
• The absence of a constant (or \( b = 0 \)) indicates that the line passes through the origin \((0,0)\).

In this exercise, the difference quotient simplifies directly to the slope of the line, indicating that the slope, or the derivative, of this linear function is constant, confirming the straight, unchanging nature of linear functions. They are crucial tools for understanding proportional relationships and are the foundation of more complex functions in calculus.