Problem 102 - Applied Calculus

Question

True or False: If \(f(x)=m x+b,\) then \(f(x+h)=f(x)+m h\).

Step-by-Step Solution

Answer

True.

Step 1: Interpret the Function

Given the function \( f(x) = m x + b \), where \( m \) is the slope and \( b \) is the y-intercept of the line. Here, the function represents a straight line.

Step 2: Substitute into the Function

Substitute \( x + h \) into the function: \( f(x + h) = m(x + h) + b \).

Step 3: Simplify the Expression

Simplify \( f(x + h) = m(x + h) + b \) to get \( f(x + h) = mx + mh + b \).

Step 4: Compare with Original Function

Now, compare \( f(x + h) = mx + mh + b \) with \( f(x) = mx + b \). When you add \( mh \) to \( f(x) \), you get \( f(x) + mh = mx + b + mh \), which is equal to \( f(x + h) \).

Step 5: Determine the Truth Value

Since \( f(x + h) = f(x) + mh \) holds true after substitution and simplification, the statement is True.

Key Concepts

Function Evaluation

Function evaluation focuses on finding the value of a function for a given input. In the context of the exercise, we consider a linear function represented by the equation \( f(x) = mx + b \). Here:

• \( f(x) \) is the output or value of the function at a particular \( x \).
• \( m \) is the slope, dictating the line's steepness.
• \( b \) is the y-intercept, marking where the line crosses the y-axis.

By substituting different values into the function, we can calculate the function's value at those points. For example, in the problem, we substitute \( x + h \) into the function. This operation allows us to see how the function behaves when the input changes by \( h \), thereby evaluating \( f(x + h) \). Understanding function evaluation is crucial, as it helps us explore the interplay between inputs and outputs and predict values under different conditions.

Slope-Intercept Form

In linear equations, the slope-intercept form is expressed as \( y = mx + b \). This form is fundamental in understanding linear functions because:

• \( m \) signifies the slope of the line, showing how much \( y \) changes for a unit change in \( x \).
• \( b \) represents the y-intercept, the point where the line crosses the y-axis.

This format gives a clear visualization of how the line behaves. The slope \( m \) reveals the direction and angle of inclination. A positive \( m \) means the line ascends, while a negative \( m \) signifies descent. Learning about the slope-intercept form aids in interpreting and sketching linear equations efficiently. In our example, this form is used to determine how changes in \( x \) and \( x + h \) affect the function’s outcome.

Substitution in Algebra

Substitution is a method in algebra used to evaluate functions or solve equations by replacing variables with numbers or other expressions. In this particular exercise, substitution is applied by replacing \( x \) with \( x + h \) in the linear function \( f(x) = mx + b \). Here's how it works:

• Substitute \( x + h \) into the function: \( f(x + h) = m(x + h) + b \).
• Simplify the resulting expression: \( f(x + h) = mx + mh + b \).

This process helps determine how the function behaves with modified inputs. Substitution allows us to simplify and manipulate expressions, facilitating deeper understanding and solution of algebraic problems. Mastering substitution can streamline solving equations and enhance comprehension of how functions operate with varying inputs.