Problem 70 - Elementary and Intermediate Algebra

Question

Let \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Find each of the following. $$ (g \circ f)\left(\frac{1}{10}\right) $$

Step-by-Step Solution

Answer

\((g \circ f)\left(\frac{1}{10}\right) = \frac{1}{100}\).

Step 1: Understand the Composition

The expression \((g \circ f)(x)\) represents the composition of two functions \(g(x)\) and \(f(x)\). It means we first apply \(f\) to \(x\) and then apply \(g\) to the result of \(f(x)\).

Step 2: Evaluate \(f\left(\frac{1}{10}\right)\)

Given the function \(f(x) = \frac{1}{x}\), substitute \(x = \frac{1}{10}\).\[ f\left(\frac{1}{10}\right) = \frac{1}{\frac{1}{10}} = 10 \]

Step 3: Evaluate \(g(10)\)

Now use the result from Step 2 as the input for \(g(x)\). The function \(g(x) = \frac{1}{x^2}\), so substitute \(x = 10\).\[ g(10) = \frac{1}{10^2} = \frac{1}{100} \]

Step 4: Conclusion: Compose and Evaluate

The composition \((g \circ f)\left(\frac{1}{10}\right)\) is evaluated by first using \(f\) and then \(g\) on the result. Therefore, \((g \circ f)\left(\frac{1}{10}\right) = \frac{1}{100}\).

Key Concepts

Inverse Functions

Inverse functions are a fascinating topic in mathematics. They are essentially functions that "undo" each other.
For example, if you have a function that converts Fahrenheit temperatures to Celsius, the inverse would convert Celsius back to Fahrenheit.
This means if you start with a temperature in Fahrenheit, convert it to Celsius, and then apply the inverse function, you would get back your original Fahrenheit temperature.Mathematically, for a function to have an inverse, it must be bijective (both injective and surjective):

• Injective: Different inputs produce different outputs. This ensures the function is one-to-one.
• Surjective: Every possible output can be obtained from some input. This ensures the function covers the entire possible set of outputs.

For instance, the function \(f(x) = \frac{1}{x}\) is its own inverse. This means applying it twice returns you to your starting point, except where the function is undefined, such as at zero.

Nested Functions

Function composition is similar to nesting in that it allows functions to be "nested" within each other. It is like placing one function inside another, which means you first compute the output of one function and then use that as the input for the next function.
The notation \((g \circ f)(x)\) indicates you first evaluate \(f(x)\), and then use that result to evaluate \(g\). This is akin to following a recipe: you perform step one, which is using \(f\), and then step two, which is applying \(g\) to the result of the first.

• Function \(f(x)\) might transform an input \(x\).
• Then, \(g\) takes \(f(x)\) and transforms it again.

In our example, the given \(f(x) = \frac{1}{x}\) and \(g(x) = \frac{1}{x^2}\) were composed to find \((g \circ f)(\frac{1}{10})\), ultimately simplifying nested processes into one calculation.

Rational Functions

Rational functions are functions that can be expressed as the quotient of two polynomials.
They have variable expressions in both the numerator and the denominator, like our example functions:

• \(f(x) = \frac{1}{x}\) - This is a simple rational function where the polynomial in the numerator is just 1.
• \(g(x) = \frac{1}{x^2}\) - Another rational function with a polynomial \(x^2\) in the denominator.

Rational functions can have discontinuities where the denominator equals zero since division by zero is undefined.
This is important for understanding any potential domain restrictions. As seen in our functions, for instance, we couldn't evaluate \(f extnormal{ or } g\) at zero.
In application, knowing the behavior of rational functions helps us predict graph shapes, asymptotes, and intercepts, all critical for solid understanding in algebra.