Problem 35 - College Algebra

Question

In Exercises \(33-44\), find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=3 x+7$$

Step-by-Step Solution

Answer

The simplified difference quotient of the function \(f(x) = 3x + 7\) is \(3\).

Step 1: Substitute the function into the difference quotient formula

Firstly, substitute \(f(x) = 3x + 7\) into the difference quotient formula \(\frac{f(x+h)-f(x)}{h}\). So it becomes: \(\frac{f(x+h)-f(x)}{h} = \frac{(3(x+h) + 7) - (3x+7)}{h}\).

Step 2: Simplify the difference of two functions in the numerator

After applying the function for \(f(x+h)\) and \(f(x)\), simplify the numerator by removing parentheses and combining like terms. The formula turns out to be: \(\frac{3x+3h+7-(3x+7)}{h} = \frac{3h}{h}\).

Step 3: Simplify by cancelling out the common factor

In this step, simplify the expression by cancelling the common factor, which in this case is \(h\), in the numerator and the denominator. That leaves us with the final simplified difference quotient of \(3\).

Key Concepts

Functions

Functions are like machines in mathematics. For every input, they produce exactly one output. In essence, a function, such as \(f(x) = 3x + 7\), transforms an input \(x\) into an output by applying a specific rule. In this example, the function is linear. Its rule is to multiply the input by 3 and then add 7.

Understanding functions helps in seeing how variables relate to each other. For every \(x\) you choose, \(f(x)\) will give you a result. Functions are incredibly useful in various mathematical fields. They describe relationships or changes between variables. In calculus, the difference quotient is a tool that examines how a function changes as its input changes. This is the first step into understanding the derivative, a key concept that tells us how fast something is changing.

• Each input \(x\) has its own output \(f(x)\).
• Linear functions like \(3x+7\) form a straight line when graphed.

Grasping functions is crucial, as they appear in algebra, calculus, and many other mathematical areas. They're the backbone for most equations and models you'll encounter.

Simplification

Simplification in mathematics is about making expressions as simple as possible. It's like tidying up your room: every term and symbol has its place, and you remove what's unnecessary.

In the given problem, we simplify the difference quotient \(\frac{3h}{h}\) to just \(3\). Here’s how simplification generally works: you break down complex expressions and then cancel or combine like terms.

• Identify like terms: These are terms that have the same variables raised to the same power. For example, \(3h\) and \(h\) can be simplified because they share the same variable \(h\).
• Cancel common factors: In fractions, if a factor appears in both the numerator and the denominator, you can cancel it out as long as it's not zero. Thus, in \(\frac{3h}{h}\), we cancel \(h\) on both ends.

Simplification clarifies expressions and makes it easier to understand their behavior and solutions. This skill is fundamental for solving algebraic expressions efficiently.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operations (like addition and multiplication). In \(f(x) = 3x + 7\), we're dealing with an algebraic expression.

Understanding these expressions involves learning how to read and manipulate them. Algebraic expressions can represent real-world problems and situations. The term \(3x\) suggests that the variable \(x\) is multiplied by 3, and adding 7 shifts the result. These expressions are foundational in algebra.

• Operations within the expression must follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Variables represent unknown values that we might want to solve for or evaluate.

Dealing with algebraic expressions can range from simplifying them, as in our difference quotient problem, to solving equations and inequalities. They're everywhere in math, from elementary education to advanced calculus. Mastering them can open doors to understanding more complex math topics.