Problem 19 - Contemporary Precalculus

Question

Find \((g \circ f)(3),(f \circ g)(1),\) and \((f \circ f)(0)\) $$f(x)=x, \quad g(x)=-3$$

Step-by-Step Solution

Answer

Answer: The compositions of the functions are as follows: - \((g \circ f)(3) = -9\) - \((f \circ g)(1) = -3\) - \((f \circ f)(0) = 0\)

Step 1: Find \(g \circ f(x)\)

To find \(g \circ f(x)\), we will substitute \(f(x)\) as the input to \(g(x)\). This means we will replace all the instances of \(x\) in \(g(x)\) with the \(f(x)\) function: $$g(f(x)) = -3f(x)$$ Now, we substitute the function \(f(x)\) with its expression \(x\): $$g(f(x)) = -3x$$

Step 2: Find \((g \circ f)(3)\)

To find \((g \circ f)(3)\), we plug in the value \(x=3\) into the \(g \circ f(x)\) function we just found: $$(g \circ f)(3) = -3(3) = -9$$

Step 3: Find \(f \circ g(x)\)

To find \(f \circ g(x)\), we will substitute \(g(x)\) as the input to \(f(x)\). This means we will replace all the instances of \(x\) in \(f(x)\) with the \(g(x)\) function: $$f(g(x)) = g(x)$$ Now, we substitute the function \(g(x)\) with its expression \(-3\): $$f(g(x)) = -3$$

Step 4: Find \((f \circ g)(1)\)

Since \(f \circ g(x) = -3\) for any value of \(x\), \((f \circ g)(1)\) is equal to \(-3\): $$(f \circ g)(1)= -3$$

Step 5: Find \(f \circ f(x)\)

To find \(f \circ f(x)\), we will substitute \(f(x)\) as the input to itself: $$f(f(x)) = f(x)$$ Now, we substitute the function \(f(x)\) with its expression \(x\): $$f(f(x)) = x$$

Step 6: Find \((f \circ f)(0)\)

To find \((f \circ f)(0)\), we plug in the value \(x=0\) into the \(f \circ f(x)\) function we just found: $$(f \circ f)(0) = 0$$ The final answers are \((g \circ f)(3)=-9\), \((f \circ g)(1)=-3\), and \((f \circ f)(0)=0\).

Key Concepts

Composite Functions

Composite functions are an essential aspect of mathematics that involves creating a new function by combining two existing functions. The idea behind composite functions is much like a series of processes where the output of one process becomes the input for another. In mathematics, this is expressed as

• if you have two functions, say \( f(x) \) and \( g(x) \),
• their composite function is denoted as \( (g \circ f)(x) \).

This means you first apply the function \( f(x) \), and then take the resulting output and use it as the input for \( g(x) \). A practical example was provided in the exercise, where:

• \( f(x) = x \)
• and \( g(x) = -3 \).

The composite function \( g \circ f \) was

• calculated by substituting \( f(x) \) into \( g(x) \)
• resulting in \( g(f(x)) = -3x \).

Understanding how composite functions work is fundamental to grasping more complex mathematical concepts.

Mathematical Functions

Mathematical functions are like machines that take an input and provide a specific output based on a rule. Every time you input the same number into a function, you should get the same output. Functions can serve various purposes, like showing relationships between variables or modeling real-world phenomena.

In the context of the given exercise:

• \( f(x) = x \) represents an identity function, which simply returns the same input as the output.
• \( g(x) = -3 \) represents a constant function, always returning the value -3 regardless of the input.

By studying these functions separately, they illustrate how they transform inputs and how this transformation follows specific rules that apply consistently. This regularity is what makes functions so indispensable in various mathematical analyses and applications.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. It's like plugging a number into a machine to see what it gives you. In dealing with mathematical functions, we focus on determining the exact outcome when a particular value is substituted into the function.

The exercise demonstrated different ways to evaluate functions:

• For \( g \circ f(3) \), the function \( g(f(x)) = -3x \) was evaluated by substituting \( x = 3 \) to find the result \(-9\).
• Similarly, for \( f \circ g(x) \), which is constant for any \( x \), the result was always \(-3\), including when \( x = 1 \).
• Lastly, the self-composition \( f \circ f(0) \) evaluated at \( x = 0 \) simply gave an output of 0, as expected from the identity function property.

Each evaluation step involves directly applying the function's rule to the specified input, ensuring precision in solving mathematical problems.