Problem 44 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=\frac{1}{x}+x\) and \(g(x)=\frac{2 x}{x^{2}+1} .\) Evaluate the following expressions. (a) \(g(f(2))\) (b) \(f(g(2))\)

Step-by-Step Solution

Answer

The values of \(g(f(2))\) and \(f(g(2))\) are 0.8 and 2, respectively.

Step 1: Evaluate f(2)

We start with function \(f\) defined as \(f(x)=\frac{1}{x}+x\). We substitute \(x=2\) into the equation to get: \(f(2)=\frac{1}{2}+2 = 2.5\)

Step 2: Evaluate g(f(2))

Next, we have function \(g\) defined as \(g(x)=\frac{2 x}{x^{2}+1}\). We have found that \(f(2) = 2.5\) from step 1. We substitute \(x = 2.5\) into function \(g\) to get: \(g(f(2)) = g(2.5) = \frac{5}{2.5^{2} + 1} = 0.8\)

Step 3: Evaluate g(2)

Now, we use function \(g\) and substitute \(x = 2\) to get: \(g(2) = \frac{4}{2^{2}+1} = 0.8\)

Step 4: Evaluate f(g(2))

Finally, we evaluate function \(f\) at \(x = g(2)\), where \(g(2) = 0.8\) as found from step 3. We substitute \(x = 0.8\) into \(f(x)\) to get: \(f(g(2))=f(0.8)=\frac{1}{0.8}+0.8 = 2\)

Key Concepts

Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics. They include addition, subtraction, multiplication, and division. In this particular exercise, you encounter these operations when evaluating functions like \( f(x) = \frac{1}{x} + x \) and \( g(x) = \frac{2x}{x^2 + 1} \).

When dealing with such expressions, each part needs careful calculation. For example, to find \( f(2) \), we need to add \( \frac{1}{2} \) and \( 2 \), which results in \( 2.5 \). Similarly, calculating \( g(2) \) involves dividing \( 4 \) by \( 5 \) to achieve \( 0.8 \).

• Make sure to handle fractions properly when dividing.
• Recollect basic arithmetic laws such as the commutative and associative properties to simplify expressions.
• Maintain accuracy by confirming each step's result before using it in further calculations.

Fractional Functions

Fractional functions involve fractions as part of their expressions. They often incorporate polynomials in their numerators and/or denominators, as seen here with \( f(x) = \frac{1}{x} + x \) and \( g(x) = \frac{2x}{x^2 + 1} \).

Mixing these fractions with other polynomial parts requires methodical calculation to simplify for exact results. For \( g(2.5) \), the function \( g(x) \) requires you to compute the fraction \( \frac{5}{2.5^2 + 1} \), giving \( 0.8 \). Be sure to:

• Determine the numerator and denominator separately before the division.
• Look out for canceling potential to simplify the fractions.
• Watch your decimal and fraction conversions closely to maintain accuracy.

Substitution Method

The substitution method is crucial for solving complex composite function problems like \( g(f(2)) \) and \( f(g(2)) \). It involves replacing a variable in one function with another value. This value could be another function's output as learned from previous steps.

In the problem, identify the output from one function (for example, \( f(2) = 2.5 \)) before feeding it into another (e.g., \( g(x) \)) to find \( g(f(2)) \).

• Start with finding each inner function's value step-by-step.
• Ensure each substitution uses accurate values as calculated.
• Confirm the final values by double-checking each filled expression for errors.