Problem 75 - College Algebra

Question

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\). $$f(x)=\frac{1}{x+2}, g(x)=4 x+3$$

Step-by-Step Solution

Answer

\(f(g(0)) = \frac{1}{5}\), \(g(f(0)) = 5\).

Step 1: Evaluate g(0)

To find \(g(0)\), substitute 0 into the function \(g(x)\). Therefore, \(g(0) = 4(0) + 3 = 3\).

Step 2: Evaluate f(g(0))

Now, substitute \(g(0) = 3\) into \(f(x)\). Thus, \(f(g(0)) = f(3) = \frac{1}{3+2} = \frac{1}{5}\).

Step 3: Evaluate f(0)

Substitute 0 into the function \(f(x)\). Therefore, \(f(0) = \frac{1}{0+2} = \frac{1}{2}\).

Step 4: Evaluate g(f(0))

Substitute \(f(0) = \frac{1}{2}\) into \(g(x)\). Thus, \(g(f(0)) = g\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right) + 3 = 2 + 3 = 5\).

Key Concepts

Algebraic Functions

Algebraic functions involve expressions that use operations like addition, subtraction, multiplication, division, and exponentiation. These functions can take many forms, such as linear, quadratic, and rational functions. In our problem, we have two algebraic functions:

• Function \(f(x) = \frac{1}{x+2}\) is a rational function because it involves division where \(x\) is part of the denominator.
• Function \(g(x) = 4x + 3\) is a linear function, characterized by degree one and having a straight-line graph.

Understanding the nature of these functions helps in figuring out how they behave when combined or when specific values are substituted into them. Rational functions, for instance, will have restrictions on domain due to division by zero, while linear functions are defined for all real numbers.

Function Evaluation

Function evaluation is the process of determining the output of a function for a given input. This involves replacing the variable in the function with a specified number. In the given exercise, you evaluate both functions by plugging in specific values:

• To evaluate \(g(0)\), substitute 0 into \(g(x)\), yielding \(g(0) = 4(0) + 3 = 3\).
• Evaluate \(f(0)\) by substituting 0 into \(f(x)\), which results in \(f(0) = \frac{1}{2}\).

This task is essential in understanding how specific inputs affect the output and helps you become adept at handling various types of functions, whether they are linear, quadratic, or more complex.

Composite Functions

Composite functions involve the combination of two functions, where the output of one function becomes the input of another. It's written as \(f(g(x))\) or \(g(f(x))\), which means applying one function to the result of another. In the exercise, you encounter both types:

• For \(f(g(0))\), you first find \(g(0)\) and then use its result in \(f(x)\). As calculated, \(f(g(0)) = f(3) = \frac{1}{5}\).
• In \(g(f(0))\), calculate \(f(0)\) first and then apply \(g\) to this result, which gives \(g(f(0)) = 5\).

Understanding composite functions is crucial, as they demonstrate how multiple functions interact to produce new results. They are useful in more complex calculations and modeling real-world scenarios where multi-step processes are involved.