Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-20: Find (a) \((f \circ g)(x)\) (b) \((g \circ f)(x)\) (c) \(f(g(-2))\) (d) \(g(f(3))\) $$ f(x)=|x|, \quad g(x)=-7 $$

Step-by-Step Solution

Answer

(a) 7, (b) -7, (c) 7, (d) -7.

Step 1: Understanding the Functions

We are given two functions: \(f(x) = |x|\) and \(g(x) = -7\). The function \(f(x)\) takes the absolute value of \(x\) and \(g(x)\) is a constant function returning \(-7\) for any \(x\).

Step 2: Compute \((f \circ g)(x)\)

The composition \((f \circ g)(x)\) means applying \(g(x)\) first and then \(f(x)\). Since \(g(x) = -7\) for any \(x\), we substitute this into \(f\):\((f \circ g)(x) = f(g(x)) = f(-7) = |-7| = 7\).

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

The composition \((g \circ f)(x)\) means applying \(f(x)\) first and then \(g(x)\). Since \(f(x) = |x|\), we substitute this into \(g\):\((g \circ f)(x) = g(f(x)) = g(|x|) = -7\) (because \(g(x)\) is constant \(-7\)).

Step 4: Compute \(f(g(-2))\)

Evaluate \(g(-2)\) first, which equals \(-7\). Then plug this value into \(f\):\(f(g(-2)) = f(-7) = |-7| = 7\).

Step 5: Compute \(g(f(3))\)

Evaluate \(f(3)\) first, which equals \(|3| = 3\). Then plug this value into \(g\):\(g(f(3)) = g(3) = -7\).

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. For instance, both \(-7\) and \(7\) have an absolute value of \(7\), since both are seven units away from zero. When we take the absolute value of any number, the result is always non-negative. This is because the absolute value essentially strips any negative sign off the number. This concept is symbolized as \(|x|\), where \(x\) can be any real number.

Consider some examples:

• \(|5| = 5\)
• \(|-3| = 3\)
• \(|0| = 0\)

The absolute value function is useful in various mathematical applications, including solving equations and analyzing real-world situations. In our context, we see it applied through the function \(f(x) = |x|\), where the function \(f\) gives back the absolute value of its input.

Constant Function

A constant function is a simple yet important type of function in mathematics. With a constant function, the output value does not change no matter what the input is. It is always the same constant. For the function \(g(x) = -7\), it tells us that regardless of the input \(x\), the function will always return \(-7\).

• For \(x = 1\), \(g(1) = -7\)
• For \(x = 0\), \(g(0) = -7\)
• For \(x = -5\), \(g(-5) = -7\)

In mathematical notation, this is expressed as a straight horizontal line on a graph, showing that the function's value does not vary with changes in \(x\). This characteristic makes constant functions particularly predictable and widely used in situations that require fixed values.

Evaluation of Functions

The evaluation of functions involves substituting specific input values into a function to obtain a corresponding output. This process helps understand the behavior of functions under varying conditions of input. For example, to evaluate \(f(x) = |x|\) at \(x = 3\), we calculate \(|3|\), which equals \(3\).

When you have composite functions like \((f \circ g)(x)\) or \((g \circ f)(x)\), it involves taking the output of one function and using it as the input for the other.

To evaluate these, you:

• First, find the output of the inner function.
• Then use this result as the input for the outer function.

As seen in our examples,

• \(f(g(-2))\) first finds \(g(-2)\), which is \(-7\), and then calculates \(f(-7)\), yielding \(7\).
• \(g(f(3))\) first finds \(f(3)\), which is \(3\), and then calculates \(g(3)\), yielding \(-7\).

This approach is crucial for understanding function compositions and their effects on values, a fundamental concept in calculus and algebra.