Problem 74 - 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. $$ f(x)=-3 x+4 $$

Step-by-Step Solution

Answer

a) \(f(x+h) = -3x - 3h + 4\) b) Difference quotient is \(-3\).

Step 1: Write Down Given Function

The given function is a linear function: \( f(x) = -3x + 4 \).

Step 2: Find the New Expression \( f(x+h) \)

To find \( f(x+h) \), substitute \( x+h \) into the function in place of \( x \). Thus, \( f(x+h) = -3(x+h) + 4 \).

Step 3: Simplify \( f(x+h) \)

Distribute the \(-3\) across \( (x+h) \) in the expression: \( f(x+h) = -3x - 3h + 4 \).

Step 4: Set Up the Difference Quotient

The difference quotient is given by:\[\frac{f(x+h) - f(x)}{h}.\]Substitute \( f(x+h) = -3x - 3h + 4 \) and \( f(x) = -3x + 4 \).

Step 5: Calculate the Numerator of the Difference Quotient

Subtract \( f(x) \) from \( f(x+h) \):\[(-3x - 3h + 4) - (-3x + 4) = -3x - 3h + 4 + 3x - 4 = -3h.\]

Step 6: Simplify the Difference Quotient

Divide the numerator \(-3h\) by \(h\):\[\frac{-3h}{h} = -3.\]Ensure \( h eq 0 \).

Step 7: Write the Final Simplified Difference Quotient

The simplified difference quotient for the given function is \(-3\).

Key Concepts

Linear Function

A linear function is one of the simplest types of functions in mathematics. It is represented by a straight line when plotted on a graph. The general formula for a linear function is:\( f(x) = mx + b \)where \( m \) is the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis. In this particular exercise, the function provided is \( f(x) = -3x + 4 \). This means that the slope of the line is \(-3\), indicating a negative slope that falls as it moves from left to right. The y-intercept of this line is \(4\), meaning the line will intersect the y-axis at the point \( (0, 4) \). Linear functions are fundamental in algebra and are used to model relationships that have a constant rate of change.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with. In this exercise, simplifying is crucial when finding \( f(x+h) \). Initially, we substitute \( x+h \) into the function, leading to an expression that can be expanded:- Start with \( f(x+h) = -3(x+h) + 4 \)- Distribute \(-3\) over the terms inside the parentheses: -\(-3 \cdot x = -3x\) -\(-3 \cdot h = -3h\)- Combine these with the constant 4:So, \( f(x+h) = -3x - 3h + 4 \).The process involves applying basic distributive properties and combining like terms. This simplification is essential for determining the difference quotient efficiently.

Substitution Method

The substitution method is a fundamental technique often used to simplify mathematical expressions and solve equations. In this exercise, substitution is used to find \( f(x+h) \). Here's how it works:- Start with the function, \( f(x) = -3x + 4 \).- Replace or substitute every instance of \( x \) with \( x+h \).So, by substituting, you get \( f(x+h) = -3(x+h) + 4 \). This step requires you to carefully place the expression \( x+h \) in place of each \( x \) in the function. This method is extremely helpful when calculating the difference quotient, which involves computing \( \frac{f(x+h) - f(x)}{h} \) and requires these substitutions to factor terms correctly. Substitution lays the groundwork for further simplification and solving.