Problem 47 - 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) \(f(f(2))\) (b) \(g(g(-1))\)

Step-by-Step Solution

Answer

The evaluation of the expressions are \(f(f(2))=\frac{29}{10}\) and \(g(g(-1))=-1\).

Step 1: Evaluate f(f(2))

First, evaluate the function f at x=2, that is \(f(2) =\frac{1}{2}+2=\frac{5}{2}\). Then substitute \(\frac{5}{2}\) into function f again: \(f\left(\frac{5}{2}\right) =\frac{2}{5}+\frac{5}{2}=\frac{29}{10}\)

Step 2: Evaluate g(g(-1))

Similarly, evaluate the function g at x=-1, that is \(g(-1)=\frac{-2}{2}=-1\). Then substitute -1 into function g again: \(g(-1)=\frac{-2}{2}=-1\).

Step 3: Simplify and Conclude Final Results

After calculating, we find that \(f(f(2))=\frac{29}{10}\) and \(g(g(-1))=-1\). Thus, these are the evaluations of the given expressions.

Key Concepts

Function Composition

Function composition is like a math matryoshka doll, where one function snuggles comfortably inside another. Think of it as a two-step math dance: you take one function and plug it into another. In the case of our problem, we have two functions, f(x) and g(x), getting cozy with each other. For instance, when we say f(f(2)), it's like asking 'What's the output of f(x) when the input is not just x, but another f(x) where x equals 2?' Just remember, it's crucial to evaluate the inner function first before feeding its result to the outer function.

To become a pro at function composition, practice by nesting different functions within each other and feel the thrill as complexity builds up. It's like a function party, and everyone's invited!

Evaluating Functions

Evaluating functions is the bread and butter of working with mathematical expressions. When we evaluate a function, such as f(x) for a specific value, we're substituting that number everywhere we see x in the function's formula. It's like a mini treasure hunt where x marks the spot, and your prize is finding out what f(x) equals. In our exercise, evaluating f(2) means we're replacing every x in f(x) = 1/x + x with 2.

As you get more familiar with evaluating, it will become second nature like playing your favorite tune on an instrument. You'll see the pattern, understand the rhythm, and play it perfectly every time.

Rational Functions

Rational functions might sound highbrow, but they're just fractions where both the numerator and the denominator are polynomials. It's like a mathematical sandwich: polynomials on the top and bottom with a division sign in the middle. In the exercise, g(x) = 2x / (x^2 + 1) is a rational function. It's a delicate balance between the top and bottom, and just like in any good sandwich, you don't want the bottom (denominator) to be zero because that would be like biting into nothingness — a mathematical no-no!

Understanding rational functions is a key ingredient in calculus and helps you navigate the curvy lines that these functions graph out. They might have twists, turns, and asymptotes, hiding like invisible walls, but once you know where they are, you'll steer clear of trouble.

Calculus Problems

Calculus problems are like puzzles waiting to be solved. With our two functions, f(x) and g(x), what seemed at first like a simple substitution game unfolds into a rich mathematical journey. From the instant you scribble down your first f(2), you're embarking on an exploration into continuous change — the heartland of calculus.

Whether it's the slopes of tangent lines, the area under curves with integration, or the intricate dance of limits, calculus provides us with tools that peel back the layers of complexity. And even though working through these problems can sometimes feel like you're scaling a sheer cliff face, remember: as you climb higher, the view gets even better. So keep pushing through those tough problems, and you'll soon be rewarded with the beauty and elegance of mathematical understanding.