Problem 38 - Introductory Algebra for College Students

Question

Find and simplify. \(\frac{f(x)-f(h)}{x-h}\) $$f(x)=8 x+9$$

Step-by-Step Solution

Answer

Therefore, the simplified form of the given expression according to the given function is 8.

Step 1: Substituting the function

First, substitute the given function \(f(x) = 8x + 9\) into the difference quotient to get: \[\frac{(8x+9) - (8h + 9)}{x - h}\]

Step 2: Simplify the Numerator

Now simplify the numerator of the fraction. The '+9' from '(8x + 9)' cancels out with the '-9' from '- (8h + 9)', so that leaves (8x - 8h) on the numerator.

Step 3: Simplify the denominator

There's nothing to simplify in the denominator as it is already in its simplest form, which is (x - h).

Step 4: Factor out the common factor

Now, in the numerator 8 is a common factor, so factor it out. We have: \[\frac{8(x - h)}{x - h}\]

Step 5: Cancel out the common terms

Now, cancel out the common term '+ (x - h)' in the numerator and the denominator. This leaves us with a constant: \[8\]

Key Concepts

Algebraic Expression Simplification

Simplifying algebraic expressions is a crucial skill in mathematics that helps you rewrite expressions in a more concise form. In the given exercise, you start by substituting the function into the expression. This involves combining like terms or reducing terms that cancel each other out.
To simplify difference quotient:
- Combine the like terms and keep track of positive and negative signs. - This means adding or subtracting the coefficients (the numbers in front of the variables). In the expression \((8x + 9) - (8h + 9)\), the \(+9\) and \(-9\) cancel each other out since they have opposite signs. Always look for such opportunities to make the expression simpler!
This leaves us with \(8x - 8h\), which is more straightforward and easier to work with than the original expression.

Function Substitution

Function substitution is substituting one expression or a part of an expression with another. When dealing with functions, you're often given an equation like \(f(x) = 8x + 9\). Substituting means plugging in specific values in place of function variables, or as seen in exercise, calculating a difference quotient.
For instance:

• The expression \((8x+9) - (8h + 9)\) is derived by substituting into the difference quotient.
• This involves placing \(f(x)\) in place of \(f(x)\) and \(f(h) = 8h + 9\) for \(f(h)\).

Be patient with substitution and ensure you follow through systematically with each step to maintain the correct variables in your outcomes. This skill is versatile and repeatedly used in calculus and algebra, providing the foundation for evaluating and analyzing functions.

Factoring

Factoring is a technique used to simplify expressions involving common multipliers. When you see terms that can be factorized, it's a cue to make your algebra work easier.
In the example \(8x - 8h\), notice that both terms share a common factor, which is 8.

• Step entails "factoring out" this common multiplier so that the expression becomes 8(x-h).
• This condensation leads to easier management of the expression and sets you up for simplifying algebraic fractions.

Factoring is not only about reducing size, but it’s about preparation for subsequent operations – especially like canceling terms, as we will see in algebraic fractions. Keep an eye on common factors as this can greatly ease the process.

Algebraic Fractions

An algebraic fraction is a fraction in which both the numerator and the denominator are algebraic expressions. Simplifying these fractions often involves factoring and canceling out common terms - much like seen in the solution process for the given problem.
Steps in simplifying include:

• Ensuring the numerator and denominator are simplified to their simplest forms.
• In the case \(\frac{8(x-h)}{x-h}\), once the factor \((x-h)\) is canceled from numerator and denominator, it simplifies to 8.

The canceling process is crucial because it extracts the expression's essence and often yields constants or simpler forms, enabling easier computations in subsequent calculations. Remember to cancel only terms that are completely multiplied, meaning having no addition or subtraction signs that can disturb straightforward cancelations.