Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

1\. For \(f(x)=x+3\) and \(g(x)=x^{2}\), find each value. (a) \((f+g)(2)\) (b) \((f \cdot g)(0)\) (c) \((g / f)(3)\) (d) \((f \circ g)(1)\) (e) \((g \circ f)(1)\) (f) \((g \circ f)(-8)\)

Step-by-Step Solution

Answer

(a) 9, (b) 0, (c) 3/2, (d) 4, (e) 16, (f) 25.

Step 1: Understanding the Functions

We begin by understanding the given functions: \(f(x) = x + 3\) and \(g(x) = x^2\). These will be used in performing various operations as specified in the questions.

Step 2: Calculate (f+g)(2)

To find \((f+g)(2)\), we first find \(f(2)\) and \(g(2)\). \(f(2) = 2 + 3 = 5\) and \(g(2) = 2^2 = 4\). Then, \((f+g)(2) = f(2) + g(2) = 5 + 4 = 9\).

Step 3: Calculate (f⋅g)(0)

To find \((f \cdot g)(0)\), calculate \(f(0)\) and \(g(0)\). \(f(0) = 0 + 3 = 3\) and \(g(0) = 0^2 = 0\). Thus, \((f \cdot g)(0) = f(0) \cdot g(0) = 3 \cdot 0 = 0\).

Step 4: Calculate (g/f)(3)

To find \((g/f)(3)\), find \(g(3)\) and \(f(3)\). \(g(3) = 3^2 = 9\) and \(f(3) = 3 + 3 = 6\). Thus, \((g/f)(3) = \frac{g(3)}{f(3)} = \frac{9}{6} = \frac{3}{2}\).

Step 5: Calculate (f∘g)(1)

To find \((f \circ g)(1)\), we first find \(g(1)\) and then \(f(g(1))\). \(g(1) = 1^2 = 1\), so \(f(g(1)) = f(1) = 1 + 3 = 4\).

Step 6: Calculate (g∘f)(1)

To find \((g \circ f)(1)\), we first find \(f(1)\) and then \(g(f(1))\). \(f(1) = 1 + 3 = 4\), so \(g(f(1)) = g(4) = 4^2 = 16\).

Step 7: Calculate (g∘f)(-8)

To find \((g \circ f)(-8)\), we first find \(f(-8)\) and then \(g(f(-8))\). \(f(-8) = -8 + 3 = -5\), so \(g(f(-8)) = g(-5) = (-5)^2 = 25\).

Key Concepts

Composite Functions

Composite functions involve combining two functions in a manner that one function's result becomes the input for another function. Essentially, you're chaining functions together. For example, if you have two functions, say \(f(x)\) and \(g(x)\), a composite function can be written as \((f \circ g)(x)\). This means you first apply \(g\) to \(x\), then apply \(f\) to the result of \(g(x)\).
For our specific exercise:

• To find \((f \circ g)(1)\), compute \(g(1)\) first, getting \(1\). Then, plug this into \(f\) to get \(f(1) = 4\).
• Similarly, for \((g \circ f)(1)\), determine \(f(1)\) first, which is \(4\), and then \(g(4)\), which results in \(16\).

Understanding the order of operations is crucial here, as switching the order changes the outcome of the composite function.

Function Addition

Function addition is a straightforward yet powerful operation. When adding two functions, you're essentially combining their outputs for given inputs. For example, with functions \(f(x)\) and \(g(x)\), the sum \((f+g)(x)\) is found by adding \(f(x) + g(x)\).
In the example provided, to find \((f+g)(2)\), you:

• Calculate \(f(2) = 5\) because \(2+3 = 5\).
• Calculate \(g(2) = 4\) because \(2^2 = 4\).
• Add these results: \(5 + 4 = 9\).

By performing these simple calculations, we combine the two functions into a new single result that reflects both inputs.

Function Multiplication

When multiplying functions, you're taking the outputs of two functions and multiplying them together for given inputs. If you have functions \(f(x)\) and \(g(x)\), multiplying them forms \((f \cdot g)(x)\).
For instance, in our exercise, to determine \((f \cdot g)(0)\), we:

• Find \(f(0) = 3\), as \(0 + 3 = 3\).
• Find \(g(0) = 0\), because \(0^2 = 0\).
• Multiply the results: \(3 \cdot 0 = 0\).

The multiplication operation shows how combining function outputs can magnify or nullify results, depending on the values involved.

Function Division

Dividing functions involves taking one function's output and dividing it by another's. Suppose you have \(f(x)\) and \(g(x)\); the division \((g / f)(x)\) calculates \(\frac{g(x)}{f(x)}\). It's important to ensure that the denominator is not zero, as division by zero is undefined.
In the example problem, for \((g / f)(3)\), you:

• Find \(g(3) = 9\), since \(3^2 = 9\).
• Find \(f(3) = 6\), because \(3 + 3 = 6\).
• Compute \(\frac{9}{6} = \frac{3}{2}\).

Function division highlights how operations can transform relationships between variables, and careful handling of inputs, especially ensuring the function does not equal zero, is critical for accurate results.