Problem 73 - College Algebra with Corequisite Support

Question

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\). \(f(x)=5 x+7, \quad g(x)=4-2 x^{2}\)

Step-by-Step Solution

Answer

\(f(g(0)) = 27\) and \(g(f(0)) = -94\).

Step 1: Find g(0)

First, we need to find the value of the function \(g(x)\) when \(x = 0\). Substitute 0 for \(x\) in the equation \(g(x) = 4 - 2x^2\):\[g(0) = 4 - 2(0)^2 = 4\]So, \(g(0) = 4\).

Step 2: Calculate f(g(0))

Now use the result from Step 1 to find \(f(g(0))\). Since \(g(0)\) is 4, substitute 4 into the function \(f(x) = 5x + 7\):\[f(g(0)) = f(4) = 5(4) + 7 = 20 + 7 = 27\]Therefore, \(f(g(0)) = 27\).

Step 3: Find f(0)

Next, find the value of the function \(f(x)\) when \(x = 0\). Substitute 0 for \(x\) in the equation \(f(x) = 5x + 7\):\[f(0) = 5(0) + 7 = 7\]So, \(f(0) = 7\).

Step 4: Calculate g(f(0))

Now use the result from Step 3 to find \(g(f(0))\). Since \(f(0)\) is 7, substitute 7 into the function \(g(x) = 4 - 2x^2\):\[g(f(0)) = g(7) = 4 - 2(7)^2 = 4 - 2(49) = 4 - 98 = -94\]Therefore, \(g(f(0)) = -94\).

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions that involve variables, numbers, and operations such as addition, subtraction, multiplication, and division. In our exercise, we deal with two algebraic functions:

• For the function \(f(x) = 5x + 7\), this is a linear function. It involves multiplying the variable \(x\) by 5 and then adding 7.
• The function \(g(x) = 4 - 2x^2\) is a quadratic function. Here, the variable \(x\) is squared and multiplied by -2, followed by adding 4.

Understanding the structure of these functions is key to solving related problems. Linear functions have a constant rate of change, while quadratic functions have a parabolic shape. Both types of functions play crucial roles in algebra.

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input value. This involves substituting the input value into the function in place of the variable. Let’s see how this applies to the functions \(f\) and \(g\).- To find \(g(0)\), substitute the value 0 in place of \(x\) in the function \(g(x) = 4 - 2x^2\). We perform the operations to get \(g(0) = 4\).- Similarly, to find \(f(0)\), substitute the value 0 in \(f(x) = 5x + 7\), resulting in \(f(0) = 7\).By evaluating these functions, we gain the necessary results for further computation, such as determining compositions of functions.

Step-by-Step Solution

A step-by-step solution helps break down complex problems into manageable parts, ensuring each stage is thoroughly understood before proceeding. Here’s how this approach is used in our function composition problem:1. **Evaluate \(g(0)\):** Begin with the inner function \(g(0)\) to get \(4\).2. **Calculate \(f(g(0))\):** Use the result from above to evaluate the outer function \(f(4)\), resulting in \(27\).3. **Evaluate \(f(0)\):** Move to the second part by finding \(f(0) = 7\).4. **Calculate \(g(f(0))\):** Finally, use the value from step 3 to find \(g(7)\), giving \(-94\).Following each step in sequence ensures clarity and allows us to verify our work at each stage. This methodical approach is not only useful for solving this problem but for tackling a wide range of algebraic exercises.