Problem 25 - A Graphical Approach to Precalculus with Limits

Question

For each pair of fimetions, (a) find ( \(f+g)(x),(f-g)(x),\) and \((f g)(x) ;\) (b) give the domains of the functions in part (a); (c) find \(\frac{t}{x}\) and give its domain; (d) find \(f \circ g\) and give is 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

All functions are defined for all real numbers except division, which excludes zero.

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

To find \((f+g)(x)\), add the functions \(f(x)\) and \(g(x)\) together. This gives: \((f+g)(x) = |x+3| + 2x\).

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

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). This gives: \((f-g)(x) = |x+3| - 2x\).

Step 3: Find (f·g)(x)

To find \((f \cdot g)(x)\), multiply \(f(x)\) and \(g(x)\) together. This gives: \((f \cdot g)(x) = |x+3| \cdot 2x \).

Step 4: Determine domains for (a)

The domain of \(f(x) = |x+3|\) is all real numbers since absolute value is defined for any real number. The domain of \(g(x) = 2x\) is also all real numbers. Therefore, the domains for \((f+g)(x)\), \((f-g)(x)\), and \((f \cdot g)(x)\) are all real numbers, \((-\infty, \infty)\).

Step 5: Find \(\frac{f}{g}(x)\)

To find \(\frac{f}{g}(x)\), divide \(f(x)\) by \(g(x)\). This gives: \(\frac{f}{g}(x) = \frac{|x+3|}{2x}\).

Step 6: Determine domain for \(\frac{f}{g}\)

The division is undefined when the denominator is zero. Therefore, solve for \(2x = 0\), which gives \(x = 0\). Thus, the domain for \(\frac{f}{g}(x)\) is all real numbers except zero, \((-\infty, 0) \cup (0, \infty)\).

Step 7: Find f ∘ g

To find \(f \circ g\), substitute \(g(x)\) into \(f(x)\): \(f(g(x)) = |2x+3|\).

Step 8: Determine domain for f ∘ g

Since \(g(x) = 2x\) and \(f(x) = |x+3|\) are both defined for all real numbers, \(f \circ g\) is defined for all real numbers, \((-\infty, \infty)\).

Step 9: Find g ∘ f

To find \(g \circ f\), substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = 2|x+3|\).

Step 10: Determine domain for g ∘ f

Since \(g(x) = 2x\) is defined for all real numbers and \(|x+3|\) is also defined for all real numbers, \(g \circ f\) is defined for all real numbers, \((-\infty, \infty)\).

Key Concepts

Domain of Functions

Understanding the domain of functions is crucial because it tells us what values can be input into a function. For any function, the domain is the set of all possible input values (usually denoted as "x") that will not lead to any undefined expressions.
In the given exercise, we see various functions involving operations such as addition, subtraction, multiplication, and division of two functions: \(f(x) = |x+3|\) and \(g(x) = 2x\).

• The absolute value function \(|x+3|\) accepts all real numbers because absolute value is defined for any real number without any issue.
• The linear function \(2x\) also accepts all real numbers since there's no restriction on "x" causing the function to become undefined.
• For composite functions such as \((f + g)(x)\), \((f - g)(x)\), and \((f \cdot g)(x)\), the domain remains all real numbers because each of these operations combines functions whose individual domains already accept all real input values.

The only situation where we encounter a restriction is in division, namely \(\frac{f}{g}(x)\). Division by zero is undefined, so we need to exclude any "x" making the denominator zero. Here, \(\frac{|x+3|}{2x}\) is undefined for \(x = 0\), giving a domain of \((-\infty, 0) \cup (0, \infty)\).
Determining domains ensures functions are used correctly, avoiding undefined scenarios during operations.

Composition of Functions

The composition of functions involves taking the output of one function and using it as the input for another function. This operation can create very complex relationships, but breaking it down step-by-step simplifies the process.
For this exercise, we have two main compositions to analyze:

• Finding \(f \circ g\): To obtain \(f \circ g\), replace every "x" in \(f(x)\) with \(g(x)\), so it becomes \(f(g(x)) = |2x+3|\). This shows the transformation due to \(g(x)\) applied to the formula for \(f(x)\). Every insertion respects the domain of both \(f(x)\) and \(g(x)\).
• Finding \(g \circ f\): Similarly, for \(g \circ f\), replace "x" in \(g(x)\) with \(f(x)\), resulting in \(g(f(x)) = 2|x+3|\). Again, we use the output of \(f(x)\) as input to \(g(x)\), adhering to the domains where both original functions are valid.

These compositions are special because they mirror the way functions can transform data, allowing us to chain operations in a meaningful and computational sequence. The final domain for these composed functions is all real numbers, \((-\infty, \infty)\), because the function operations don't introduce any new restrictions compared to the individual function domains.

Absolute Value Function

The absolute value function is one of the simplest non-linear functions and plays a critical role in various mathematical contexts. Represented as \(|x|\), it simply measures the distance of a number from zero on the number line, always resulting in a non-negative value.
In the context of the function \(f(x) = |x+3|\), the concept is slightly shifted by adding 3 inside the absolute value. This means that instead of measuring the distance from zero, it measures the distance from -3. Therefore, any input \(x\) to \(f(x)\) first shifts left by 3 units, then evaluates the absolute value.
Some characteristics of this function include:

• Domain: Covers all real numbers because absolute values don't possess any intrinsic restrictions.
• Range: Outputs will always be zero or greater.
• Symmetry: This function mirrors over the y-axis whenever the expression inside the absolute value is purely linear (like \(x+3\)).

Understanding the behavior of the absolute value function helps in dissecting entire transformations of the terms inside the bars, crucial when integrating into larger expressions or multiple function operations.