Problem 14 - Calculus with Concepts in Calculus

Question

Let \(f(x)=\frac{x-1}{x^{2}+1}\) and \(g(x)=x^{1 / 4}\). Find the specified values. $$ g(f(1)) $$

Step-by-Step Solution

Answer

The value of \( g(f(1)) \) is 0.

Step 1: Evaluate f(1)

To find \( f(1) \), substitute \( x = 1 \) into the function \( f(x) = \frac{x - 1}{x^2 + 1} \). This gives \( f(1) = \frac{1 - 1}{1^2 + 1} = \frac{0}{2} = 0 \). So, \( f(1) = 0 \).

Step 2: Evaluate g(f(1))

Now substitute \( f(1) = 0 \) into the function \( g(x) = x^{1/4} \). This gives \( g(f(1)) = g(0) = 0^{1/4} = 0 \).

Key Concepts

Composite Functions

Composite functions involve combining two or more functions, where the output of one function becomes the input of another. In our problem, we are dealing with functions \( f(x) \) and \( g(x) \). The aim is to evaluate \( g(f(x)) \), which means "apply \( f(x) \) first, and then use its result in \( g(x) \)." This is written as \( (g \circ f)(x) \) or simply \( g(f(x)) \).
Composite functions are a crucial part of understanding how functions interact. They allow us to establish complex relationships and transformations using simpler, individual functions. The order of operations is essential when dealing with composite functions, as reversing the order can yield different results. For instance, \( g(f(x)) \) is generally not the same as \( f(g(x)) \).
Let's summarize the process:

• Evaluate the inner function \( f(x) \) at a given value.
• Use the result as input for the outer function \( g(x) \).
• The outcome is \( g(f(x)) \).

Using composite functions allows us to solve complex problems step-by-step, ensuring that each part of the function chain is correctly evaluated.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying mathematical expressions to solve problems more efficiently. In this example, we perform algebraic manipulation to find \( f(1) \) first, before using its output in \( g(x) \).
Let's break down the manipulation:

• Substitute \( x = 1 \) in \( f(x) = \frac{x - 1}{x^2 + 1} \).
• This simplifies to \( \frac{1 - 1}{1^2 + 1} \), which equals \( \frac{0}{2} = 0 \).
• Using clear algebraic steps ensures accuracy and clarity.

Algebraic manipulation may appear simple, but it's fundamental in solving function-based problems like these. When dealing with fractions, it’s important to conduct operations in a clear, step-by-step manner, as errors can easily propagate. Practicing precise algebraic manipulation builds a strong foundation for handling more complex algebraic scenarios.

Step-by-Step Solution

Solving math problems in a step-by-step manner is a powerful strategy to ensure accuracy and understanding. This approach involves breaking down complex problems into simple, manageable steps.
In our exercise, here’s the step-by-step breakdown:

• First, identify \( f(1) \) by substituting in the function \( f(x) \).
• Simplify \( f(1) \) to find the result, in this case, 0.
• Next, substitute this result into \( g(x) \), which gives \( g(0) \).
• Simplify \( g(0) \) to get the final answer, which is 0.

This method allows students to focus on individual tasks rather than tackling the entire problem at once. It’s a methodical way of thinking that not only provides solutions but also gives insight into the relationships between functions and how changes in variables affect these functions.
In essence, a step-by-step solution is not only about finding the answer but also about understanding the entire journey towards reaching the answer.