Problem 32 - Intermediate Algebra

Question

Determine the indicated functional values. (Objective 2 ) If \(f(x)=\frac{2}{x-1}\) and \(g(x)=-\frac{3}{x}\), find \((f \circ g)(1)\) and \((g \circ f)(-1) .\)

Step-by-Step Solution

Answer

(f ∘ g)(1) = -1/2 and (g ∘ f)(-1) = 3.

Step 1: Understand Function Composition

First, we need to understand we are asked to find the compositions \( (f \circ g)(x) = f(g(x)) \) and \( (g \circ f)(x) = g(f(x)) \). We'll evaluate these compositions at specific values \( x = 1 \) and \( x = -1 \).

Step 2: Compute \( g(1) \)

Substitute \( x = 1 \) into \( g(x) = -\frac{3}{x} \). We have: \[ g(1) = -\frac{3}{1} = -3 \].

Step 3: Compute \( (f \circ g)(1) \)

Now that we know \( g(1) = -3 \), compute \( f(-3) \). Substitute \( x = -3 \) into \( f(x) = \frac{2}{x-1} \):\[ f(-3) = \frac{2}{-3-1} = \frac{2}{-4} = -\frac{1}{2} \].Hence, \((f \circ g)(1) = -\frac{1}{2}\).

Step 4: Compute \( f(-1) \)

Substitute \( x = -1 \) into \( f(x) = \frac{2}{x-1} \). We have:\[ f(-1) = \frac{2}{-1-1} = \frac{2}{-2} = -1 \].

Step 5: Compute \( (g \circ f)(-1) \)

Now that we know \( f(-1) = -1 \), compute \( g(-1) \). Substitute \( x = -1 \) into \( g(x) = -\frac{3}{x} \):\[ g(-1) = -\frac{3}{-1} = 3 \].Hence, \((g \circ f)(-1) = 3\).

Key Concepts

Functional Values

When dealing with functions, we often need to find functional values. A functional value is the result you get when you plug a specific number into a function. For instance, if you have a function like \( f(x) = \frac{2}{x-1} \), and you want to know what happens when \( x = 2 \), you simply substitute the value of \( x \) into the function to obtain \( f(2) = \frac{2}{2-1} = 2 \).
This is known as evaluating the function at a specific value. Functional values provide us with specific outputs of functions, making it easier to understand how a function behaves at particular points.

Function Evaluation

Function evaluation is a straightforward process. It involves substituting a given number into a function to obtain an output. Suppose you have two functions: \( f(x) = \frac{2}{x-1} \) and \( g(x) = -\frac{3}{x} \). To evaluate \( f(x) \) at \( x = 3 \), replace \( x \) with \( 3 \) in the function:
\[ f(3) = \frac{2}{3-1} = \frac{2}{2} = 1 \].
Similarly, evaluating \( g(x) \) at \( x = 5 \) involves:
\[ g(5) = -\frac{3}{5} \].
These substitutions are crucial in finding out how a function acts at any chosen value. By evaluating functions, we can explore their behavior better and predict future functional outputs.

Composition of Functions

Complex mathematical problems often involve the composition of functions. Function composition involves combining two functions in such a way that the output of one becomes the input of another. Mathematically, if you have two functions \( f(x) \) and \( g(x) \), the composition \( (f \circ g)(x) \) means applying \( g \) first and then \( f \).
For example, if \( f(x) = \frac{2}{x-1} \) and \( g(x) = -\frac{3}{x} \), finding \( (f \circ g)(1) \) involves these steps:

• First, evaluate \( g(1) \): \( g(1) = -\frac{3}{1} = -3 \)
• Then substitute \( g(1) \) into \( f \): \( f(-3) = \frac{2}{-3-1} = -\frac{1}{2} \)

Thus, \( (f \circ g)(1) = -\frac{1}{2} \).
Function composition allows us to build complex functions from simpler ones, opening up possibilities for further mathematical exploration and understanding.