Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-14: If \(a\) is a positive real number, find (a) \(g\left(\frac{1}{a}\right)\) (b) \(\frac{1}{g(a)}\) (c) \(g(\sqrt{a})\) (d) \(\sqrt{g(a)}\) $$ g(x)=\frac{x^{2}}{x+1} $$

Step-by-Step Solution

Answer

(a) \( \frac{1}{a(1+a)} \); (b) \( \frac{a+1}{a^2} \); (c) \( \frac{a}{\sqrt{a} + 1} \); (d) \( \frac{a}{\sqrt{a+1}} \)

Step 1: Evaluate g(1/a)

To find \( g\left(\frac{1}{a}\right) \), substitute \( \frac{1}{a} \) into the function \( g(x) = \frac{x^2}{x+1} \).\[g\left(\frac{1}{a}\right) = \frac{\left(\frac{1}{a}\right)^2}{\frac{1}{a} + 1} = \frac{\frac{1}{a^2}}{\frac{1+a}{a}} = \frac{1}{a^2} \cdot \frac{a}{1+a} = \frac{1}{a(1+a)}\]

Step 2: Evaluate 1/g(a)

Find \( \frac{1}{g(a)} \) where \( g(a) = \frac{a^2}{a+1} \).\[\frac{1}{g(a)} = \frac{1}{\frac{a^2}{a+1}} = \frac{a+1}{a^2}\]

Step 3: Evaluate g(sqrt(a))

Substitute \( \sqrt{a} \) into the function to find \( g(\sqrt{a}) \).\[g(\sqrt{a}) = \frac{(\sqrt{a})^2}{\sqrt{a} + 1} = \frac{a}{\sqrt{a} + 1}\]

Step 4: Evaluate sqrt(g(a))

To evaluate \( \sqrt{g(a)} \), take the square root of \( g(a) = \frac{a^2}{a+1} \).\[\sqrt{g(a)} = \sqrt{\frac{a^2}{a+1}} = \frac{\sqrt{a^2}}{\sqrt{a+1}} = \frac{a}{\sqrt{a+1}}\]

Key Concepts

Understanding Real Numbers

Real numbers are a fundamental concept in mathematics, encompassing all the numbers we use in everyday life. These include:

• Integers like -3, 0, or 5
• Rational numbers, such as 1/2 or 3.75
• Irrational numbers, which can't be expressed as fractions, like \(\pi\) or \(\sqrt{2}\)

Real numbers are crucial because they form the basis for defining more advanced mathematical structures, like algebraic expressions and functions. In the context of the given exercise, we are particularly focused on positive real numbers. This means all the numbers greater than zero, excluding zero itself, which are used as inputs in our function evaluations.

Decomposing Algebraic Expressions

An algebraic expression is a combination of constants, variables, and the algebraic operations: addition, subtraction, multiplication, division, and exponentiation. For example, the expression \(x^2 + 3x + 2\) is algebraic because it contains:

• The variable \(x\)
• Coefficients (numerical constants) 2 and 3
• Operations like addition and exponentiation

To evaluate such expressions at a specific point means substituting the variable with a number (often a real number) and simplifying. In our context, the function \(g(x) = \frac{x^2}{x+1}\) is a type of algebraic expression known as a rational function because it can be expressed as the quotient of two polynomials.

Exploring Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. The function \(g(x) = \frac{x^2}{x+1}\) is rational because:

• The numerator, \(x^2\), is a polynomial.
• The denominator, \(x+1\), is also a polynomial.

Rational functions are significant in mathematics due to their diverse applications in real-world scenarios and their interesting properties, such as vertical and horizontal asymptotes. When evaluating a rational function like \(g(x)\) at a particular point, it's essential to ensure the denominator doesn't zero out, which could lead to an undefined expression. This is especially important to remember when handling complex substitutions like \(g\left(\frac{1}{a}\right)\) or \(g(\sqrt{a})\) in the exercise, where the calculations can quickly become intricate.