Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, use each set of functions to find \(f(g(h(x)))\). Simplify your answers. 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

(a) \(g(f(x)) = \frac{x^2}{2x^2 - 2x + 1}\), (b) \(g(f(2)) = \frac{4}{5}\).

Step 1: Understand the Composition of Functions

To solve this problem, we need to understand function composition. Given two functions, \(f\) and \(g\), the composition \((g \circ f)(x)\) means the same as \(g(f(x))\). We will insert the output from \(f(x)\) into the function \(g\).

Step 2: Find \(f(x)\)

From the problem, we have \(f(x)=\frac{1-x}{x}\). This is the function whose output we will plug into function \(g\).

Step 3: Evaluate \(g(f(x))\)

We will now substitute \(f(x)\) into \(g(x)\). Since \(g(x) = \frac{1}{1+x^2}\), \(g(f(x)) = \frac{1}{1+(\frac{1-x}{x})^2}\).

Step 4: Simplify the Expression \((1-x)/x)^2\)

Calculate \(\left(\frac{1-x}{x}\right)^2 = \frac{(1-x)^2}{x^2}\).

Step 5: Substitute the Expression into \(g(x)\)

Replace the squared term in \(g(f(x))\) with the expression found in Step 4: \(g(f(x)) = \frac{1}{1+\frac{(1-x)^2}{x^2}}\).

Step 6: Simplify the Denominator

To simplify further, multiply the denominator and numerator by \(x^2\) to eliminate the fraction: \(g(f(x)) = \frac{x^2}{x^2 + (1-x)^2}\).

Step 7: Simplify \(x^2 + (1-x)^2\)

Expand and combine \((1-x)^2\): \((1-x)^2 = 1 - 2x + x^2\). Thus, \(x^2 + 1 - 2x + x^2 = 2x^2 - 2x + 1\).

Step 8: Final Simplified Form for \((g \circ f)(x)\)

Substitute back into the expression: \(g(f(x)) = \frac{x^2}{2x^2 - 2x + 1}\). This is the simplified form for \((g \circ f)(x)\).

Step 9: Evaluate \((g \circ f)(2)\)

Substitute \(x = 2\) into the simplified expression: \((g \circ f)(2) = \frac{2^2}{2(2)^2 - 2(2) + 1} = \frac{4}{8 - 4 + 1} = \frac{4}{5}\).

Key Concepts

Algebraic Functions

Algebraic functions are a key part of many math problems. They involve using symbols and letters to represent numbers and operations. This allows us to write mathematical expressions more generally. Here's what you need to know:

• An algebraic function can include addition, subtraction, multiplication, division, and even roots.
• For example, given an algebraic function like \(f(x) = \frac{1-x}{x}\), we use \(x\) as a variable that can take different values.
• The output depends on the specific value of \(x\) you plug into the function.

When dealing with algebraic functions, think of the function as a machine. You feed it an input and it gives you an output. This is crucial when working with function compositions, as you'll see in the context of function composition problems.

Simplifying Expressions

Simplifying expressions means making them easier to understand or solve without changing their value. This is particularly useful in mathematics when dealing with complex expressions.

• Let's take the expression \(\frac{1}{1+\left(\frac{1-x}{x}\right)^2}\) from our function composition problem.
• The goal is to break down the expression into a simpler form, usually by combining like terms and reducing fractions.
• In this example, \(\left(\frac{1-x}{x}\right)^2\) is expanded to \(\frac{(1-x)^2}{x^2}\). We then focus on simplifying this further.

By multiplying the entire numerator and denominator by \(x^2\), we remove the fraction within a fraction, resulting in \(\frac{x^2}{2x^2 - 2x + 1}\). Such steps make it easier to evaluate or solve expressions.

Evaluating Functions

Evaluating functions involves cutting through the layers of a function and finding the output once a specific input is plugged in. This often follows the simplification of expressions to make calculation smoother.

• Function evaluation typically comes at the final step, once you've got a simplified expression for your function composition.
• For instance, after simplifying \( (g \circ f)(x) = \frac{x^2}{2x^2 - 2x + 1}\), we substitute a specific value of \(x\), such as \(x = 2\).
• By substituting \(x = 2\), the expression becomes \(\frac{4}{5}\), which shows the specific output for this input.

Understanding and applying the process of evaluating functions helps you confirm your work. It's an essential skill for solving real-world math problems efficiently.