Problem 74 - College Algebra

Question

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\). $$f(x)=\sqrt{x+4}, g(x)=12-x^{3}$$

Step-by-Step Solution

Answer

\( f(g(0)) = 4 \) and \( g(f(0)) = 4 \).

Step 1: Evaluate g(0)

Begin by finding the value of the function \( g(x) \) at \( x = 0 \). Substitute \( x = 0 \) into the function: \[ g(0) = 12 - (0)^3 = 12 \]. Thus, \( g(0) = 12 \).

Step 2: Evaluate f(g(0))

Now that we have \( g(0) = 12 \), substitute \( 12 \) into function \( f(x) \): \[ f(g(0)) = f(12) = \sqrt{12 + 4} = \sqrt{16} = 4 \]. Thus, \( f(g(0)) = 4 \).

Step 3: Evaluate f(0)

Find the value of the function \( f(x) \) at \( x = 0 \). Substitute \( x = 0 \) into the function: \[ f(0) = \sqrt{0 + 4} = \sqrt{4} = 2 \]. Thus, \( f(0) = 2 \).

Step 4: Evaluate g(f(0))

Now that we have \( f(0) = 2 \), substitute \( 2 \) into function \( g(x) \): \[ g(f(0)) = g(2) = 12 - (2)^3 = 12 - 8 = 4 \]. Thus, \( g(f(0)) = 4 \).

Key Concepts

Evaluating Functions

Evaluating functions is one of the fundamental skills in algebra that allows you to determine the output of a function for a given input. A function essentially maps each input to exactly one output. To evaluate a function like \( f(x) = \sqrt{x+4} \), simply substitute the input value that you are interested in into the function in place of \( x \). For example, for \( f(0) \), substitute 0 into the function:

• Start with \( f(x) = \sqrt{x+4} \)
• Replace \( x \) with 0, giving \( f(0) = \sqrt{0+4} \)
• Calculate the result, \( f(0) = \sqrt{4} = 2 \)

This process helps in understanding how changes in inputs affect the outputs and allows you to solve various types of function problems.

Function Operations

Function operations involve performing arithmetic operations with functions such as addition, subtraction, multiplication, division, and composition. In our exercise, we are focusing on function composition, which is combining two functions so that the output of one function becomes the input of another. This is expressed as \((f \circ g)(x) = f(g(x))\).

When we computed \( f(g(0)) \), we were performing a composition of the two functions \( f \) and \( g \).

• First, evaluate \( g(0) \) to get the output from the \( g \) function: \( g(0) = 12 - 0^3 = 12 \)
• Then, use this output as the input for \( f \), calculating \( f(12) = \sqrt{12+4} = \sqrt{16} = 4 \)

Function composition is powerful because it allows you to merge different processes and operations, often simplifying complex problems into more manageable steps.

Algebraic Manipulation

Algebraic manipulation is crucial for solving equations, simplifying expressions, and evaluating functions. It involves rearranging and simplifying functions or expressions to make them easier to work with. In the given problem, algebraic manipulation helps in evaluating the outputs correctly.

Let's see how algebraic manipulation is applied in different steps here:

• Evaluating \( f(0) = \sqrt{0+4} = \sqrt{4} = 2 \) involved substituting \( x = 0 \) and simplifying the expression under the square root.
• For \( g(2) = 12 - (2)^3 \), substitute \( x = 2 \) and simplify the cube and subtraction: \(12 - 8 = 4\).

These manipulations ensure accuracy and provide a clear path from setup to solution. Mastering these skills makes tackling algebraic problems more straightforward and ensures clarity in mathematical reasoning.