Problem 25 - A Graphical Approach to College Algebra

Question

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=|x+3|, g(x)=2 x$$

Step-by-Step Solution

Answer

The domains for (f+g)(x), (f-g)(x), (fg)(x), \(\frac{f}{8}(x)\), \(f \circ g\), and \(g \circ f\) are all real numbers, \((-\infty, \infty)\).

Step 1: Find (f+g)(x)

The operation \((f+g)(x)\) involves adding the two functions \(f(x)\) and \(g(x)\). \[ (f+g)(x) = f(x) + g(x) = |x+3| + 2x \]

Step 2: Find (f-g)(x)

The operation \((f-g)(x)\) involves subtracting \(g(x)\) from \(f(x)\). \[ (f-g)(x) = f(x) - g(x) = |x+3| - 2x \]

Step 3: Find (f g)(x)

The operation \((fg)(x)\) involves multiplying the two functions \(f(x)\) and \(g(x)\). \[ (fg)(x) = f(x) \cdot g(x) = |x+3| \cdot 2x = 2x|x+3| \]

Step 4: Domain of (f+g)(x), (f-g)(x), (f g)(x)

The domain of these functions depends on the domain restrictions of \(f(x)=|x+3|\) and \(g(x)=2x\). Both functions are defined for all real numbers since \(|x+3|\) is defined for all \(x\) and \(2x\) is also defined for all \(x\). Thus, the domain for \((f+g)(x)\), \((f-g)(x)\), and \((fg)(x)\) is all real numbers, \((-\infty, \infty)\).

Step 5: Find \(\frac{f}{8}\) and give its domain

To find \(\frac{f}{8}\), divide \(f(x)\) by 8: \[ \frac{f}{8}(x) = \frac{|x+3|}{8} \] The absolute value function \(|x+3|\) is defined for all \(x\), and dividing by 8 does not impose any additional restrictions. Thus, the domain of \(\frac{f}{8}(x)\) is also all real numbers, \((-\infty, \infty)\).

Step 6: Find \(f \circ g\) and give its domain

The composition \(f \circ g\) means applying \(f\) to \(g(x)\). \[ (f \circ g)(x) = f(g(x)) = f(2x) = |2x+3| \] The function \(f(2x)\) is an absolute value function which is defined for all real numbers. Thus, the domain of \(f \circ g\) is all real numbers, \((-\infty, \infty)\).

Step 7: Find \(g \circ f\) and give its domain

The composition \(g \circ f\) means applying \(g\) to \(f(x)\). \[ (g \circ f)(x) = g(f(x)) = g(|x+3|) = 2|x+3| \] The function \(2|x+3|\) is a real-valued function and is defined for all real numbers. Therefore, the domain of \(g \circ f\) is also all real numbers, \((-\infty, \infty)\).

Key Concepts

Domain of Functions

In the realm of mathematics, understanding the domain of a function is crucial. The domain refers to the set of all possible input values (or "x" values) that a function can accept without resulting in any undefined expressions or operations. For the functions discussed above, we have two functions:

• \( f(x) = |x+3| \)
• \( g(x) = 2x \)

Both of these functions are quite friendly in terms of domains. The absolute value function, \(|x+3|\), is defined for every real number. Similarly, the linear function \(2x\) can take any real number as its input. Therefore, both functions have a domain of all real numbers, expressed as \((-\infty, \infty)\).
When performing operations like addition, subtraction, or multiplication on these functions, such as \((f+g)(x), (f-g)(x), \text{or} \ (fg)(x)\), the domain remains unchanged. This is because there are no additional operations that would restrict the possible inputs further. Thus, the domain for these combined operations is also \((-\infty, \infty)\). When dividing, such as finding \(\frac{f}{8}(x)\), we again face no further constraints since 8 is a non-zero constant, preserving the domain of all real numbers.

Composition of Functions

Function composition involves applying one function to the results of another. It's denoted \(f \circ g\) or \(g \circ f\), where \(f \circ g\) means ":take \(g(x)\) and use it as the input for \(f\)." Here's how it works with the given functions:

• \(f \circ g (x) = f(g(x)) = |2x + 3|\)
• \(g \circ f (x) = g(f(x)) = 2|x + 3|\)

For the composition \(f \circ g\), we apply \(g\) first, which is multiplying by 2, resulting in \(2x\). This outcome becomes the input for \(f\), and we compute \(|2x + 3|\). Since both \(f\) and \(g\) are defined for all real numbers, the composition \(f \circ g\) is also valid for all real numbers, thus having a domain of \((-\infty, \infty)\).
In the reverse composition \(g \circ f\), we apply \(f\) first, resulting in \(|x+3|\), and then this result is multiplied by 2 in \(g\), giving us \(2|x + 3|\). Again, no new domain restrictions are introduced, so the domain of \(g \circ f\) also remains \((-\infty, \infty)\). Function composition thus broadens possibilities while retaining domain integrity when dealing with functions defined on all real numbers.

Absolute Value Function

The absolute value function, often symbolized by vertical bars \(|x|\), is a unique type of function that always produces a non-negative result. It represents the "distance" of a number \(x\) from zero on the number line, which means that \(|x|\) is always non-negative. For instance, \(|x + 3|\) shifts this before calculating how far away the result is from zero:

• For inputs where \(x + 3 \geq 0\), the output is simply \(x + 3\).
• For inputs where \(x + 3 < 0\), the output is \( -(x + 3) \).

This dual behavior essentially bends the line along the x-axis at the point it intersects the negative part, effectively "mirroring" the negative side above the x-axis.
Because the absolute value function is designed to handle all real-number inputs without exceptions, its domain naturally extends across the entire real number line, \((-\infty, \infty)\), as indicated in the previous discussions of domain. Understanding \(|x|\) and its resulting effects in algebra and calculus can yield insights into function behaviors and is invaluable for grasping more complex functions and operations.