Problem 73 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Find g(0)

To find \( f(g(0)) \), we need to evaluate \( g(0) \) first. Substitute \( x = 0 \) into the function \( g(x) \). Thus, \( g(0) = 4 - 2(0)^2 = 4 \).

Step 2: Evaluate f(g(0))

Now that we have \( g(0) = 4 \), substitute this value into the function \( f(x) \). Thus, \( f(g(0)) = f(4) = 5(4) + 7 = 20 + 7 = 27 \).

Step 3: Find f(0)

Next, evaluate \( f(0) \) as required for calculating \( g(f(0)) \). Substitute \( x = 0 \) into the function \( f(x) \). Thus, \( f(0) = 5(0) + 7 = 7 \).

Step 4: Evaluate g(f(0))

Now that we have \( f(0) = 7 \), substitute this value into the function \( g(x) \). Thus, \( g(f(0)) = g(7) = 4 - 2(7)^2 = 4 - 2(49) = 4 - 98 = -94 \).

Key Concepts

Step by Step Solution

Solving a problem step by step can make it less intimidating and help you understand each part thoroughly. Here, we'll tackle the concept of function composition using a step-by-step approach. The goal is to compute specific values such as \( f(g(0)) \) and \( g(f(0)) \) using two given functions.

• Step 1 involves evaluating \( g(0) \). You substitute \( x = 0 \) into \( g(x) = 4 - 2x^2 \). Simplifying gives \( g(0) = 4 \), as \( 2(0)^2 = 0 \).
• In Step 2, we substitute \( g(0) = 4 \) into \( f(x) = 5x + 7 \), resulting in \( f(g(0)) = f(4) = 27 \).
• Next, in Step 3, assess \( f(0) \) by substituting \( x = 0 \) into \( f(x) \), giving \( f(0) = 7 \).
• Finally, Step 4 requires inserting \( f(0) = 7 \) back into \( g(x) \), leading to \( g(f(0)) = g(7) = -94 \).

Following these steps ensures you systematically approach function composition and any related question, making complex calculations more digestible.

Evaluating Functions

Evaluating functions is a fundamental skill in algebra and calculus. It involves finding the value of a function given a specific input. Let's break it down using the functions \( f(x) = 5x + 7 \) and \( g(x) = 4 - 2x^2 \).
To evaluate \( g(0) \), replace \( x \) with \( 0 \) in the function \( g \). You compute: \( g(0) = 4 - 2(0)^2 = 4 \).
For \( f(0) \), you substitute \( x = 0 \) into \( f(x) \), obtaining: \( f(0) = 5(0) + 7 = 7 \).Evaluating functions might include:

• Substituting the input value into the function expression.
• Simplifying the mathematical operations to reach a numeric output.
• Checking the result to ensure accuracy in calculation.

This technique is essential because it lays the groundwork for understanding more complex arrangements such as composite functions. Once mastered, evaluating more complicated functions becomes effortless.

Composite Functions

Composite functions involve plugging one function into another. This merger creates a new function that's a combination of existing ones. With \( f(x) = 5x + 7 \) and \( g(x) = 4 - 2x^2 \), our task is to find expressions like \( f(g(x)) \) or \( g(f(x)) \).
For \( f(g(0)) \), we start by evaluating \( g(0) \) to get \( g(0) = 4 \) (from \( g(x) \)) and then \( f(4) \) yields \( 27 \) (using \( f(x) \)).
Meanwhile, \( g(f(0)) \) requires first evaluating \( f(0) \) to reach \( 7 \), then substituting back into \( g(x) \), resulting in \( -94 \). Here's how composite functions work:

• Begin with evaluating the inside function first, which acts like the input to the outside function.
• Take the output from the first evaluation and substitute it into the second function.
• Carry out any simplifications needed to arrive at the final result.

Composite functions can initially seem complex, but by following these systematic steps, you can master this important concept and apply it to a myriad of mathematical problems.