Problem 22 - College Algebra

Question

Given the functions \(f(x)=\frac{1-x}{x}\) and \(g(x)=\frac{1}{1+x^{2}}\) find the following: a. \((g \circ f)(x)\) b. \((g \circ f)(2)\)

Step-by-Step Solution

Answer

(g ∘ f)(x) = \frac{x^2}{2x^2 - 2x + 1}; (g ∘ f)(2) = \frac{4}{5}."

Step 1: Understand the Composition of Functions

The notation \((g \circ f)(x)\) refers to the composition of the two functions \(f(x)\) and \(g(x)\). This means we will substitute \(f(x)\) into \(g(x)\). In simpler terms, find \(g(f(x))\).

Step 2: Substitute f(x) into g(x)

Start with the inner function, which is \(f(x) = \frac{1-x}{x}\). Replace \(x\) in \(g(x) = \frac{1}{1+x^{2}}\) with \(f(x)\). Therefore, \((g \circ f)(x) = g(f(x)) = \frac{1}{1+(\frac{1-x}{x})^{2}}\).

Step 3: Simplify (g ∘ f)(x)

Let's simplify the expression inside the denominator, \(1 + (\frac{1-x}{x})^{2}\). Start with \((\frac{1-x}{x})^{2}\), which simplifies to \(\frac{(1-x)^{2}}{x^{2}}\). Therefore, \(1 + (\frac{1-x}{x})^{2} = 1 + \frac{(1-x)^{2}}{x^{2}} = \frac{x^{2} + (1-x)^{2}}{x^{2}}\).

Step 4: Expand and Simplify Further

Expand \((1-x)^{2}\) to get \(1 - 2x + x^{2}\). Substitute into the denominator: \( \frac{x^{2} + 1 - 2x + x^{2}}{x^{2}} = \frac{2x^{2} - 2x + 1}{x^{2}}\). Thus, \((g \circ f)(x) = \frac{x^{2}}{2x^{2} - 2x + 1}\).

Step 5: Solve \((g \circ f)(2)\)

Now, substitute \(x = 2\) into \((g \circ f)(x)\). This gives \((g \circ f)(2) = \frac{2^{2}}{2(2)^{2} - 2(2) + 1}\).

Step 6: Simplify \((g \circ f)(2)\)

Calculate the numerator: \(2^{2} = 4\). Now calculate the denominator: \(2(2^{2}) - 2 \times 2 + 1 = 8 - 4 + 1 = 5\). Hence, \((g \circ f)(2) = \frac{4}{5}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental in the study of mathematics. They consist of numbers, variables, and arithmetic operations. In our exercise, both functions, \(f(x)\) and \(g(x)\), are presented as algebraic expressions.
\(f(x)\) is represented by \(\frac{1-x}{x}\) while \(g(x)\) is given as \(\frac{1}{1+x^2}\). These expressions show how values are manipulated using different variables and operations.
When working with algebraic expressions, several operations can be performed:

• Substitution: Replacing a variable with a given value.
• Simplification: Reducing the expression to its simplest form.
• Evaluation: Calculating the expression by substituting variables with numbers.

Understanding algebraic expressions is essential for performing function evaluations and compositions.

Function Evaluation

Function evaluation involves calculating the value of a function for a given input. This process is critical in determining how a function behaves and what results it produces.
To evaluate a function:

• Identify the function, for example, \(f(x)\) or \(g(x)\).
• Select the input \(x\) that you wish to evaluate.
• Substitute this value into the function.

For instance, evaluating \((g \circ f)(2)\) means substituting \(x = 2\) into the composed function \((g \circ f)(x)\). This requires you to first perform the function composition, then simplify it, before performing the substitution. It's an excellent exercise to see the nature of both basic functions and their combined form.

Substitution Method

The substitution method is an essential technique in mathematics, often used in function composition and solving equations. It involves replacing a variable with a specific value or another expression.
In function composition, represented as \((g \circ f)(x)\), the substitution method allows one to combine functions by taking the output of one function and using it as the input of another. This means taking \(f(x) = \frac{1-x}{x}\) and substituting it where \(x\) appears in \(g(x) = \frac{1}{1+x^2}\). As a result, we get:
\[g(f(x)) = \frac{1}{1+\left(\frac{1-x}{x}\right)^2}\]
After the substitution, further simplification is necessary to obtain a cleaner expression, which may involve expanding and combining algebraic terms. This approach is particularly useful in breaking down complex problems into manageable steps and finding solutions efficiently.