Problem 59 - Intermediate Algebra

Question

Find \(\frac{f(a+h)-f(a)}{h}\) for each of the given functions. (Objective 4) $$f(x)=-3 x+6$$

Step-by-Step Solution

Answer

The result is \(-3\).

Step 1: Identify the Function

The given function is \( f(x) = -3x + 6 \). This is a linear function.

Step 2: Substitute \( x = a + h \) into the Function

We need to find \( f(a + h) \). Substitute \( x = a + h \) into the function: \[ f(a+h) = -3(a+h) + 6 \].

Step 3: Simplify the Expression for \( f(a + h) \)

Distribute the \(-3\) through the terms inside the parentheses:\[ f(a+h) = -3a - 3h + 6 \].

Step 4: Find \( f(a) \)

Using the original function, replace \( x \) with \( a \):\[ f(a) = -3a + 6 \].

Step 5: Formulate \( f(a+h) - f(a) \)

Subtract \( f(a) \) from \( f(a+h) \):\[ f(a+h) - f(a) = (-3a - 3h + 6) - (-3a + 6) \].

Step 6: Simplify \( f(a+h) - f(a) \)

Combine like terms in the expression:\(-3a - 3h + 6 + 3a - 6 = -3h \).

Step 7: Divide by \( h \)

To complete the expression, divide \(-3h\) by \( h \):\[ \frac{f(a+h) - f(a)}{h} = \frac{-3h}{h} \].

Step 8: Simplify the Final Expression

Cancel \( h \) in the numerator and denominator:\[ \frac{-3h}{h} = -3 \].

Key Concepts

Understanding Linear Functions

A linear function is a fundamental concept in algebra. It represents a relationship between two variables that form a straight line when graphed. For a linear function, you'll often see it expressed in the form of \( f(x) = mx + b \), where:

• \( m \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

In our example, the function is defined as \( f(x) = -3x + 6 \). This tells us a few things:

• \( m = -3 \), so the line slopes downward from left to right.
• \( b = 6 \), which means the line crosses the y-axis at the point \((0, 6)\).

Understanding the structure of a linear function helps you quickly identify how changes in \( x \) affect \( f(x) \). This is crucial when finding the difference quotient, as it’s a way to measure the slope over a small interval.

The Art of Simplification

Simplification in algebra involves making expressions easier to work with. It’s about reducing complexity while maintaining the original value. In our exercise, simplification is used several times.Let's look at the simplification of \( f(a+h) \):

• Start with \( f(a+h) = -3(a+h) + 6 \).
• Distribute the \( -3 \) to both \( a \) and \( h \), leading to \( -3a - 3h + 6 \).

This step-by-step approach makes it clear how to handle different parts of the expression. Another key simplification is in the expression \( f(a+h) - f(a) \). When combined, it simplifies to \( -3h \) due to the cancellation of terms like \( -3a + 6 \) and \( 3a - 6 \). Recognizing these like terms and canceling them out is crucial when simplifying expressions to their most basic form.

Exploring Algebraic Expressions

An algebraic expression involves numbers, variables, and operators, like addition or multiplication. Understanding them is essential for tasks like our exercise where we work with the function \( f(x) = -3x + 6 \).Breaking down expressions into understandable steps is the key:

• Identify all parts: constants, variables, and coefficients.
• Recognize operations, such as \( -3x \) means multiplying \( x \) by \(-3 \).
• Apply order of operations, making sure to handle addition and multiplication correctly.

By examining expressions like \( f(a+h) \), you become familiar with substitution, replacing \( x \) with \( a+h \) to form \( -3(a+h) + 6 \). Simplifying follows, such as distributing and combining like terms.Overall, working through algebraic expressions strengthens problem-solving skills in algebra, laying the groundwork for more complex mathematical concepts.