Problem 55 - Precalculus Essentials

Question

Find. a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=4 x-3, g(x)=5 x^{2}-2$$

Step-by-Step Solution

Answer

(f \circ g)(x) = 20x^2 - 11, (g \circ f)(x) = 80x^2 - 120x + 43, (f \circ g)(2) = 69, and (g \circ f)(2) = 123.

Step 1: Compute f(g(x))

First, substitute \( g(x) \) into the function \( f \). This gives \( f(g(x)) = 4(5x^2 - 2) - 3 = 20x^2 - 8 - 3 = 20x^2 - 11 \). This is the result of \((f \circ g)(x)\).

Step 2: Compute g(f(x))

Next, substitute \( f(x) \) into \( g \), giving \( g(f(x)) = 5(4x - 3)^2 - 2 = 5(16x^2 - 24x + 9) - 2 = 80x^2 - 120x + 45 - 2 = 80x^2 - 120x + 43 \). This is the result of \((g \circ f)(x)\).

Step 3: Evaluate (f o g)(2)

Substitute \( x = 2 \) into the equation for \((f \circ g)(x)\) to get \((f \circ g)(2) = 20*2^2 - 11 = 20*4 - 11 = 80 - 11 = 69 \).

Step 4: Evaluate (g o f)(2)

Substitute \( x = 2 \) into the equation for \((g \circ f)(x)\) to get \((g \circ f)(2) = 80*2^2 - 120*2 + 43 = 80*4 - 240 + 43 = 320 - 240 + 43 = 123 \).

Key Concepts

Function operations

When working with functions, there might be situations where you want to combine them to create a new function. Function operations allow us to do this by using processes like addition, subtraction, multiplication, division, or composition. Composition is a special type of function operation where we apply one function to the results of another function. It's like putting a puzzle piece into place to see a complete picture.
In mathematics, when we see notations like \((f \circ g)(x)\), it means we are composing function \(f\) with function \(g\). Here, \(g(x)\) is evaluated first, and then \(f\) is applied to the result of \(g(x)\). This is a powerful tool because it allows more complex operations to be represented simply and effectively. Remember, the order matters: \((f \circ g)(x)\) is generally different from \((g \circ f)(x)\).
It’s like a two-step recipe: first, use \(g\) to process \(x\), then take that result and put it in \(f\). When performing the operation \(f(g(x))\), ensure you follow these two steps closely.

• Evaluate \(g(x)\).
• Substitute this result into \(f\).

Substitution in functions

Understanding substitution in functions is key to performing function compositions successfully. Substitution allows us to replace the input of one function with another function's output. Think of substitution as filling in a blank. Imagine \(f(x)\) as a machine where you feed it a value, and it processes it to give you a new value. When substituting, you are feeding the output of \(g(x)\) into the \(f(x)\) machine.
For example, with \(f(x) = 4x - 3\) and \(g(x) = 5x^2 - 2\), to find \((f \circ g)(x)\), replace \(x\) in \(f\) with \(g(x)\). So, \(f(g(x)) = 4(5x^2 - 2) - 3\). You calculate the value by substituting the expression for \(g(x)\) into \(f\).

• Calculate \(g(x)\) to obtain an expression or number.
• Replace \(x\) in \(f(x)\) with the expression from \(g(x)\).

This systematic substitution ensures you seamlessly integrate the results of one function into another, maintaining the mathematical integrity of function composition.

Calculating function values

Calculating function values involves evaluating expressions to discover specific numbers corresponding to those functions. This step requires plugging in a particular value for \(x\) and simplifying. Let's see how this works through an example from composing functions.
Suppose we have determined that \((f \circ g)(x) = 20x^2 - 11\). To find the value at \(x = 2\), substitute \(2\) into \((f \circ g)(x)\): \[(f \circ g)(2) = 20 \cdot 2^2 - 11 = 80 - 11 = 69. \]We followed these steps:

• Write down the expression you have for the composed function.
• Substitute the given number (in this case, \(x = 2\)) into the expression.
• Simplify the equation to find the result, which measures the function's output at that specific point.

Similarly, for \((g \circ f)(x) = 80x^2 - 120x + 43\), replacing \(x\) with \(2\) and simplifying gives us: \[(g \circ f)(2) = 80 \cdot 2^2 - 120 \cdot 2 + 43 = 320 - 240 + 43 = 123.\] Mastering how to calculate these values accurately helps understand the functions' behavior at specific points and can aid in graphing them as well.