Problem 20 - Calculus for Biology and Medicine

Question

Suppose that \(f(x)=x^{4}, x \in \mathbf{R} .\) For each of the following functions \(g(x)\), determine whether \((f \circ g)(x)=(g \circ f)(x)\) or not. (a) \(g(x)=x+1, x \in \mathbf{R}\). (b) \(g(x)=\sqrt{x}, x \in \mathbf{R}\). (c) \(g(x)=\frac{1}{x}, x>0\). (d) \(g(x)=-x, x \in \mathbf{R}\). (e) \(g(x)=|x|, x \in \mathbf{R}\).

Step-by-Step Solution

Answer

(a) Not equal; (b) Equal; (c) Equal; (d) Not equal; (e) Equal.

Step 1: Determine f ◦ g(x) for (a)

For \(g(x) = x + 1\), we need to calculate \((f \circ g)(x)\). Substitute \(g(x)\) into \(f(x)\): \[(f \circ g)(x) = f(g(x)) = f(x + 1) = (x+1)^4.\]Expand \((x+1)^4\) to \[(x+1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1.\]

Step 2: Determine g ◦ f(x) for (a)

Calculate \((g \circ f)(x)\) where \(f(x) = x^4\): \[(g \circ f)(x) = g(f(x)) = g(x^4) = x^4 + 1.\]

Step 3: Compare f ◦ g(x) and g ◦ f(x) for (a)

Now compare the results from Step 1 and Step 2:\[(f \circ g)(x) = x^4 + 4x^3 + 6x^2 + 4x + 1 eq x^4 + 1 = (g \circ f)(x).\] Thus, \((f \circ g)(x) eq (g \circ f)(x)\) for \(g(x) = x + 1\).

Step 4: Determine f ◦ g(x) for (b)

For \(g(x) = \sqrt{x}\), calculate \((f \circ g)(x)\):\[(f \circ g)(x) = f(g(x)) = f(\sqrt{x}) = (\sqrt{x})^4 = x^2.\]

Step 5: Determine g ◦ f(x) for (b)

Calculate \((g \circ f)(x)\) where \(f(x) = x^4\):\[(g \circ f)(x) = g(f(x)) = g(x^4) = \sqrt{x^4} = |x^2|.\]

Step 6: Compare f ◦ g(x) and g ◦ f(x) for (b)

Now compare:\[(f \circ g)(x) = x^2 \text{ and } (g \circ f)(x) = |x^2|.\]Since \(x^2 = |x^2|\) for all real \(x\),\((f \circ g)(x) = (g \circ f)(x)\) for \(g(x) = \sqrt{x}\).

Step 7: Determine f ◦ g(x) for (c)

For \(g(x) = \frac{1}{x}\), calculate \((f \circ g)(x)\):\[(f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^4 = \frac{1}{x^4}.\]

Step 8: Determine g ◦ f(x) for (c)

Calculate \((g \circ f)(x)\) where \(f(x) = x^4\):\[(g \circ f)(x) = g(f(x)) = g(x^4) = \frac{1}{x^4}.\]

Step 9: Compare f ◦ g(x) and g ◦ f(x) for (c)

Now compare:\[(f \circ g)(x) = \frac{1}{x^4} \text{ and } (g \circ f)(x) = \frac{1}{x^4}.\]Thus, \((f \circ g)(x) = (g \circ f)(x)\) for \(g(x) = \frac{1}{x}\).

Step 10: Determine f ◦ g(x) for (d)

For \(g(x) = -x\), calculate \((f \circ g)(x)\):\[(f \circ g)(x) = f(g(x)) = f(-x) = (-x)^4 = x^4.\]

Step 11: Determine g ◦ f(x) for (d)

Calculate \((g \circ f)(x)\) where \(f(x) = x^4\):\[(g \circ f)(x) = g(f(x)) = g(x^4) = -x^4.\]

Step 12: Compare f ◦ g(x) and g ◦ f(x) for (d)

Now compare:\[(f \circ g)(x) = x^4 \text{ and } (g \circ f)(x) = -x^4.\]Thus, \((f \circ g)(x) eq (g \circ f)(x)\) for \(g(x) = -x\).

Step 13: Determine f ◦ g(x) for (e)

For \(g(x) = |x|\), calculate \((f \circ g)(x)\):\[(f \circ g)(x) = f(g(x)) = f(|x|) = (|x|)^4 = x^4.\]

Step 14: Determine g ◦ f(x) for (e)

Calculate \((g \circ f)(x)\) where \(f(x) = x^4\):\[(g \circ f)(x) = g(f(x)) = g(x^4) = |x^4|.\]

Step 15: Compare f ◦ g(x) and g ◦ f(x) for (e)

Now compare:\[(f \circ g)(x) = x^4 \text{ and } (g \circ f)(x) = |x^4|.\]Since \(x^4\) is non-negative for all real \(x\), \(|x^4| = x^4\).Thus, \((f \circ g)(x) = (g \circ f)(x)\) for \(g(x) = |x|\).

Key Concepts

Composite Functions

Understanding composite functions is like understanding how two machines work together to produce an output. When you have two functions, say \(f(x)\) and \(g(x)\), their composition \((f \circ g)(x)\) means you first apply \(g\) to \(x\), and then apply \(f\) to the result of \(g(x)\). So, \((f \circ g)(x)\) equals \(f(g(x))\). It's like having a recipe that requires you to first prepare ingredients (using \(g\)), and then cook them (using \(f\)). In contrast, \( (g \circ f)(x) \) flips that order, applying \(f\) first, then \(g\). Understanding the order of these operations is crucial in getting the desired outcome.

Function Equality

When comparing \((f \circ g)(x)\) to \((g \circ f)(x)\), you are checking if the processes of mixing ingredients and cooking lead to the same dish as doing it the other way around. Two compositions are equal if they result in the same function for every input \(x\). During our exercise, we saw that sometimes these compositions were equal, like with \(g(x) = \sqrt{x}\) and \(g(x) = \frac{1}{x}\), despite starting with different approaches. For function equality, not only should the end results match numerically, but also their process, offering insights into how the functions influence each other.

Mathematical Operations

Composition combines mathematical operations within two functions. Consider \(f(x) = x^4\) and \(g(x) = x + 1\); when you compose them, you're adding before exponentiating \(g(x)\) and exponentiating before adding \(f(x)\). This can change the numbers around significantly, affecting the end result. Let's not forget the power of combining these through other forms like \(|x|\), \(-x\), or even \(\frac{1}{x}\), all introducing their own transformations. Each operation modifies \(x\) in its own way, emphasizing the need for careful handling to maintain an accurate final result.

Problem Solving

Solving problems involving composite functions often requires breaking down the task into simpler steps. First, understand each function’s role individually. Then, determine how they interact when composed in different orders. This methodical approach was illustrated in the exercise by calculating \((f \circ g)(x)\) and \((g \circ f)(x)\) separately before comparing. For better accuracy in problem-solving, it's helpful to have a strategy:

• Identify each function and its operation.
• Calculate the composition in the needed order.
• Compare the results for equality.

These steps simplify complex functions, turning daunting problems into manageable tasks.