Problem 72 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Evaluate g(0)

To find \(f(g(0))\), we first need to evaluate \(g(0)\). Substitute \(x = 0\) into the function \(g(x) = 7 - x^2\) to get \(g(0) = 7 - 0^2 = 7\).

Step 2: Evaluate f(g(0))

Now, we need to find \(f(g(0)) = f(7)\). Using the function \(f(x) = 4x + 8\), substitute \(x = 7\) to get \(f(7) = 4(7) + 8 = 28 + 8 = 36\).

Step 3: Evaluate f(0)

To find \(g(f(0))\), we first need to evaluate \(f(0)\). Substitute \(x = 0\) into the function \(f(x) = 4x + 8\) to get \(f(0) = 4(0) + 8 = 8\).

Step 4: Evaluate g(f(0))

Now, we need to find \(g(f(0)) = g(8)\). Using the function \(g(x) = 7 - x^2\), substitute \(x = 8\) to get \(g(8) = 7 - 8^2 = 7 - 64 = -57\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function for a specific input. It's like feeding a function a number and seeing what comes out. Let's look at the function \(f(x) = 4x + 8\). To evaluate it, we replace the \(x\) in the equation with the number of interest. For instance, if we want to find \(f(0)\), we substitute \(0\) for \(x\). This gives \(f(0) = 4(0) + 8 = 8\). The result, \(8\), is what you get when 0 is plugged into the function. The same process applies to other functions like \(g(x) = 7 - x^2\). Suppose you're curious about \(g(0)\); substituting \(0\) into the function results in \(g(0) = 7 - 0^2 = 7\). It's as simple as that! Remember, function evaluation is a fundamental skill, allowing us to see how functions behave with different inputs.

Substitution Method

The substitution method is a technique used to solve complex equations by breaking them down into simpler parts. It involves replacing one variable or function in an equation with another function or value you're trying to solve for, step by step. For composite functions like \(f(g(x))\), you start by evaluating the innermost function \(g(x)\) with the given input.

• If you're given \(x = 0\), start by calculating \(g(0)\).
• Once you have \(g(0)\), take this result and substitute it back into the function \(f(x)\).

This gives you \(f(g(0))\). For example, by finding \(g(0) = 7\) first, you substitute \(7\) into \(f(x)\) to get \(f(7)\) which equals \(36\). This method makes solving composite functions much more manageable by focusing on each function individually.

Algebraic Functions

Algebraic functions are expressions involving numbers, variables, and arithmetic operations like addition, subtraction, multiplication, division, and exponentiation. They play a crucial role in mathematics, helping to describe relationships with equations. Consider \(f(x) = 4x + 8\) and \(g(x) = 7 - x^2\), which are both algebraic functions.

• \(f(x)\) is a simple linear function, representing a straight line when graphed, due to its first-degree terms.
• \(g(x)\) is a quadratic function, which forms a parabola because of the \(x^2\) term.

These equations allow you to perform various calculations and transformations, such as finding composite functions or solving equations.Understanding how to manipulate and use algebraic functions is vital in solving mathematical problems efficiently. With algebraic functions, you can model real-world scenarios and predict outcomes with accuracy.