Problem 58 - Intermediate Algebra

Question

Find \(\frac{f(a+h)-f(a)}{h}\) for each of the given functions. (Objective 4) $$f(x)=5 x-4$$

Step-by-Step Solution

Answer

The answer is 5.

Step 1: Substitute into the function

Start by substituting \(x = a + h\) into the function \(f(x) = 5x - 4\). This gives us \(f(a+h) = 5(a+h) - 4\).

Step 2: Simplify \(f(a+h)\)

Simplify the expression for \(f(a+h)\). Thus, \(f(a+h) = 5(a+h) - 4 = 5a + 5h - 4\).

Step 3: Calculate \(f(a+h) - f(a)\)

Substitute \(f(a) = 5a - 4\) into the expression and calculate \(f(a+h) - f(a) = (5a + 5h - 4) - (5a - 4)\). This simplifies to \(5h\).

Step 4: Divide by \(h\)

To find \(\frac{f(a+h) - f(a)}{h}\), divide the expression from Step 3 by \(h\). Thus, \(\frac{5h}{h} = 5\).

Key Concepts

Linear Functions

Linear functions are one of the most fundamental concepts in algebra and mathematics overall. A linear function is a function of the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. In this form, \( m \) is known as the slope and \( b \) is the y-intercept.- **Slope (\( m \))**: Determines the steepness and direction of the line. A positive slope means the line ascends as \( x \) increases, and a negative slope means it descends.- **Y-intercept (\( b \))**: The point at which the line crosses the y-axis. This is the value of \( f(x) \) when \( x = 0 \).The linear function given in the exercise is \( f(x) = 5x - 4 \).- Here, the slope \( m = 5 \), indicating that the line rises steeply as \( x \) increases.- The y-intercept \( b = -4 \) shows that the line crosses the y-axis at \( -4 \).Linear functions are the simplest kind of functions, involving only constant factors and addition or subtraction. Understanding their structure helps in many areas of mathematics and real-world applications, such as predicting trends or understanding change rates.

Function Notation

Function notation is a way to denote functions in mathematics clearly and concisely. Seeing expressions like \( f(x) \), \( f(a+h) \), and \( f(a) \) might be confusing at first, but they're simply labels for expressing outputs of functions.- **\( f(x) \)**: This notation specifies that \( f \) is a function of \( x \). The expression within the parentheses is the input of the function.- **Substitution**: When we see \( f(a+h) \), it means we're plugging \( a+h \) in place of \( x \) in the function. So for \( f(x) = 5x - 4 \), substituting \( a+h \) gives \( f(a+h) = 5(a+h) - 4 \).Function notation serves many purposes:

• It clearly indicates the function's variable.
• It allows for substitution and evaluation at specific points.
• Provides a compact way to represent and manipulate algebraic expressions.

Understanding function notation is crucial because it allows us to work with and compare different function behaviors efficiently.

Algebraic Simplification

Algebraic simplification is the process of reducing mathematical expressions to their simplest form, making them easier to work with or interpret. In the exercise, simplification plays a key role in finding the difference quotient.Here's how it works in this exercise:- **Simplifying \( f(a+h) \)**: After substituting \( a+h \) into the function, you have \( f(a+h) = 5(a+h) - 4 \). Simplifying this gives \( f(a+h) = 5a + 5h - 4 \).- **Calculating \( f(a+h) - f(a) \)**: When you subtract \( f(a) = 5a - 4 \) from \( f(a+h) \), the terms \( 5a \) and \(-4\) cancel out, leaving \( 5h \).Algebraic simplification involves:

• Combining like terms.
• Cancelling out terms.
• Making expressions more manageable.

Finally, dividing the expression \( 5h \) by \( h \) results in \( 5 \). Simplifying expressions not only provides clarity but also prepares them for further mathematical operations, like solving equations or finding limits.