Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-52: Simplify the difference quotient \(\frac{f(x)-f(a)}{x-a}\) if \(x \neq a\). \(f(x)=\sqrt{x-3}\) (Hint: Rationalize the numerator.)

Step-by-Step Solution

Answer

The simplified difference quotient is \( \frac{1}{\sqrt{x-3} + \sqrt{a-3}} \).

Step 1: Substitute the function into the difference quotient

First, substitute the function \( f(x) = \sqrt{x-3} \) and \( f(a) = \sqrt{a-3} \) into the difference quotient:\[\frac{f(x)-f(a)}{x-a} = \frac{\sqrt{x-3} - \sqrt{a-3}}{x-a}\]

Step 2: Rationalize the numerator

To rationalize the numerator, multiply both the numerator and the denominator by the conjugate of the numerator:\[\frac{\sqrt{x-3} - \sqrt{a-3}}{x-a} \times \frac{\sqrt{x-3} + \sqrt{a-3}}{\sqrt{x-3} + \sqrt{a-3}}\] This gives:\[\frac{(\sqrt{x-3})^2 - (\sqrt{a-3})^2}{(x-a)(\sqrt{x-3} + \sqrt{a-3})}\]

Step 3: Simplify the expression

Simplify the squared terms in the numerator:\[ (\sqrt{x-3})^2 = x-3 \quad \text{and} \quad (\sqrt{a-3})^2 = a-3\]Therefore, the numerator becomes:\[ x-3 - (a-3) = x - a\]Putting this back into the expression, it simplifies to:\[ \frac{x-a}{(x-a)(\sqrt{x-3} + \sqrt{a-3})}\]

Step 4: Cancel out the \((x-a)\) terms

Since \( x eq a \), you can cancel \( x-a \) from both the numerator and denominator, resulting in:\[\frac{1}{\sqrt{x-3} + \sqrt{a-3}}\]

Key Concepts

Rationalizing the Numerator

Rationalizing the numerator is a technique used to eliminate radicals from the numerator of a fraction. This can make the expression easier to work with and simplify. In problems involving square roots, like our example, we multiply by the conjugate.
The conjugate of an expression like \( a - b \) is \( a + b \). So, when the numerator is \( \sqrt{x-3} - \sqrt{a-3} \), the conjugate would be \( \sqrt{x-3} + \sqrt{a-3} \). By multiplying the fraction by this conjugate over itself (which is equivalent to 1),
you do not change the value of the expression, but you transform the numerator into a difference of squares.

• This technique relies on the identity: \((a-b)(a+b) = a^2 - b^2\).
• This often simplifies complex expressions into simpler forms without radicals.
• The resulting algebraic manipulations will be easier because we focus on subtraction and addition of rational numbers rather than handling radicals.

Rationalizing can be incredibly handy, especially in calculus and advanced algebra, as it prepares fractions for further simplification.

Simplifying Algebraic Expressions

Algebraic simplification is the process of making an expression more straightforward to work with. This often means canceling out terms, factoring,
or using identities to consolidate expressions. When dealing with fractions, as in our example, simplification is critical.
Once rationalized, the numerator becomes \((\sqrt{x-3})^2 - (\sqrt{a-3})^2 = x - a\). This allows for some clear simplification.
In our solution, after multiplying by the conjugate,

• We observed that the numerator \( x - a \) could cancel with the denominator \( x - a \).
• This required that \( x eq a \) to avoid division by zero.
• The result was a simplified expression free of the original fraction, reducing our original difference quotient to a simpler form.

This process is a valuable skill when tackling derivative calculations and algebraic manipulations, letting us focus more on results than dealing with complex numerators.

Radical Expressions

Radical expressions involve roots, such as square roots. They're common in algebra and calculus, often representing functions or solutions to equations.
In our example, \(f(x) = \sqrt{x-3}\), which involves a square root. Working with radicals can initially seem challenging given their rules.
Here's what you should remember about radicals:

• The square root function \( \sqrt{x} \) requires non-negative inputs, as negative square roots are not real numbers.
• Operations with radicals follow specific rules similar to exponents.
• Combining or separating radicals requires careful adherence to these rules to maintain equivalence.

Once you multiply expressions involving radicals appropriately, like using conjugates,
you simplify them further by removing or reducing radical terms, leading to rational expressions that are easier to interpret and handle in calculations.