Problem 72 - College Algebra

Question

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\). $$f(x)=4 x+8, g(x)=7-x^{2}$$

Step-by-Step Solution

Answer

\(f(g(0)) = 36\), \(g(f(0)) = -57\).

Step 1: Evaluate g(0)

First, find the value of \(g(0)\). Substitute 0 into the function \(g(x) = 7 - x^2\). This gives us \(g(0) = 7 - (0)^2 = 7\).

Step 2: Substitute g(0) into f

Now, substitute the value from Step 1 into the function \(f\). We need to calculate \(f(g(0))\) which is \(f(7)\). Substitute 7 into \(f(x) = 4x + 8\). This gives \(f(7) = 4(7) + 8 = 28 + 8 = 36\).

Step 3: Evaluate f(0)

Next, find the value of \(f(0)\). Substitute 0 into the function \(f(x) = 4x + 8\). This gives us \(f(0) = 4(0) + 8 = 8\).

Step 4: Substitute f(0) into g

Now, substitute the value from Step 3 into the function \(g\). We need to calculate \(g(f(0))\) which is \(g(8)\). Substitute 8 into \(g(x) = 7 - x^2\). This gives \(g(8) = 7 - (8)^2 = 7 - 64 = -57\).

Key Concepts

Algebraic Functions

Algebraic functions are expressions that involve basic arithmetic operations such as addition, subtraction, multiplication, and division, along with exponentiation. Think of them as recipes where you input a number and follow the instructions to get a result. They form a fundamental part of algebra and can be as simple as a linear function like \(f(x) = 4x + 8\) or more complex quadratic functions like \(g(x) = 7 - x^2\).

Algebraic functions can be recognized by their equation forms, often consisting of constants, variables, and arithmetic operators. In our example:

• \(f(x) = 4x + 8\) is a linear function, where \(4x\) specifies a rate of change and \(8\) is the constant term.
• \(g(x) = 7 - x^2\) is a quadratic function due to the \(x^2\) term, creating a parabolic curve.

Understanding algebraic functions is crucial as they serve as the basis for various calculations and transformations in mathematics, including the creation of composite functions.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. It is like feeding a number into a machine and seeing what comes out. To evaluate a function, you replace the variable in the function with the provided input value.

For instance, if you have \(g(x) = 7 - x^2\) and you want to find \(g(0)\), you simply substitute every \(x\) with \(0\). From there, \(g(0) = 7 - (0)^2 = 7\).

• Evaluate \(f(x) = 4x + 8\) for \(x = 0\): \(f(0) = 4(0) + 8 = 8\) – resulting in the output of 8.
• Evaluate \(g(x)\) for a different number, say 8: \(g(8) = 7 - (8)^2 = 7 - 64 = -57\).

Function evaluation is fundamental because it allows us to determine specific values and even solve real-world problems using mathematical models.

Composite Functions

Composite functions combine two or more functions, feeding the output of one function into the input of another. It is a powerful tool that extends the reach of basic algebraic functions. If you imagine each function as a separate machine, a composite function is like connecting these machines so that what comes out of the first is processed by the next.

Consider the compositions \(f(g(x))\) and \(g(f(x))\). To find these, you follow a two-step process:

• Evaluate the inner function first to get an intermediate result.
• Use this result as the input for the outer function.

In the example,
- For \(f(g(0))\), first compute \(g(0)\) to get 7, then compute \(f(7)\), resulting in 36.
- For \(g(f(0))\), find \(f(0)\) first, which is 8, and then calculate \(g(8)\), the result being -57.
Composite functions are key in both algebra and calculus for modeling multi-step processes, making them valuable for advanced learning.