Problem 104 - Brief Applied Calculus

Question

$$ \begin{array}{l} \text { True or False: If } f(x)=m x+b, \text { then }\\\ f(x+h)=f(x)+m h \end{array} $$

Step-by-Step Solution

Answer

True. Given \( f(x) = mx + b \), then \( f(x+h) = f(x) + mh \) holds true.

Step 1: Understand the Given Function

We are given the function \( f(x) = mx + b \), which is a linear function where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Calculate \( f(x + h) \)

To find \( f(x+h) \), substitute \( x + h \) into the function: \[ f(x+h) = m(x+h) + b = mx + mh + b \]

Step 3: Derive \( f(x) + mh \)

Calculate \( f(x) + mh \) using the original function: \[ f(x) + mh = (mx + b) + mh = mx + mh + b \]

Step 4: Compare \( f(x + h) \) and \( f(x) + mh \)

Both expressions are \( mx + mh + b \). Hence, \( f(x + h) = f(x) + mh \) is indeed true.

Key Concepts

Slope

The slope of a linear function is a key concept in mathematics. It's the number that tells us how steep a line is, or how much it changes in the vertical direction (up and down) for a given change in the horizontal direction (left to right). Understanding slope is crucial because it lets us know how one variable changes in relation to another. In the function \( f(x) = mx + b \), the slope is represented by \( m \). If we imagine standing on a hill, a higher slope \( m \) means the hill is steeper.

• Positive Slope: The line goes upwards as you move to the right. The larger the slope, the steeper the ascent.
• Negative Slope: The line goes downwards as you move to the right. The more negative the slope, the steeper the descent.
• Zero Slope: The line is flat, indicating no change in vertical movement.

Being able to quickly identify and understand the slope helps in graphing the function and predicting future behavior of the line.

Function Notation

Function notation is like a special code in math to represent the output of a function for a given input. It helps us work with equations more efficiently. When you see \( f(x) \), it means the function \( f \) is applied to the input \( x \). Think of it like a machine where you put something in and get something else out. So, if \( f(x) = mx + b \), then for any input \( x \), you calculate the output by multiplying \( x \) with the slope \( m \) and then adding the y-intercept \( b \). Important aspects of function notation:

• Input: The number you are substituting into the function, represented by \( x \).
• Output: The result after the function has taken the input, symbolized as \( f(x) \).

Function notation makes it easy to understand relationships between different values and to recognize patterns.

Algebraic Expressions

Algebraic expressions are like puzzles made from numbers, variables (like \( x \)), and operations (like addition and multiplication). Learning how to manipulate these expressions is a key skill in algebra. In a linear function such as \( f(x) = mx + b \), the expression combines:

• Variables: Represent unknown values, such as \( x \) in the equation.
• Constants: These are fixed values, like \( m \) (slope) and \( b \) (y-intercept) in the function.
• Operators: Symbols like \( + \) and \( - \) that indicate operations to perform.

Solving problems often involves manipulating these expressions according to the rules of arithmetic to find unknown values or simplify expressions. It's all about breaking down the problem into smaller parts and working with each part according to mathematical laws.