Problem 22 - College Algebra

Question

Determine the interval(s) on which the function is increasing and decreasing. $$a(x)=\sqrt{-x+4}$$

Step-by-Step Solution

Answer

The function is decreasing on \((-\infty, 4)\).

Step 1: Find the domain of the function

Determine the allowed values of \(x\) such that the function \(a(x) = \sqrt{-x + 4}\) is defined. Since the expression under the square root must be non-negative, solve the inequality \(-x + 4 \geq 0\). This gives \(x \leq 4\). Thus, the domain is \([-\infty, 4]\).

Step 2: Determine the derivative of the function

Find the derivative \(a'(x)\) to determine where the function \(a(x)\) is increasing or decreasing. The function is \(a(x) = (-x+4)^{1/2}\). Using the chain rule, \(a'(x) = \frac{1}{2}(-x+4)^{-1/2} \cdot (-1) = \frac{-1}{2\sqrt{-x+4}}\).

Step 3: Determine the intervals of increase or decrease

Because \(a'(x) = \frac{-1}{2\sqrt{-x+4}}\), observe that \(a'(x) < 0\) for all \(x\) in its domain \((-\infty, 4)\), which means the function is always decreasing on this interval.

Step 4: Check behavior at the endpoint

The endpoint \(x = 4\) is included in the domain. At \(x = 4\), the derivative does not exist because \(\sqrt{0}\) in the denominator makes it undefined, but this point does not affect the function's decreasing nature before this endpoint.

Key Concepts

Domain of a Function

To understand the behavior of a function, we first need to know its domain. The domain represents all the possible input values (usually x-values) for which the function is defined.
For the function given in the exercise, \( a(x) = \sqrt{-x + 4} \), the domain is determined by the condition that the expression inside the square root must be non-negative. This is because square roots of negative numbers are not defined in the set of real numbers.
Thus, we solve the inequality:

• \(-x + 4 \geq 0\)

Solving this, we find \(x \leq 4\). Therefore, the domain of the function is all x-values from negative infinity up to 4, written as \((-\infty, 4]\). The endpoint, 4, is included because at \(x = 4\), the expression inside the root becomes zero, which is valid.

Derivative

The derivative of a function gives us important information about the rate of change of the function's values. It essentially tells us how the function's output changes as its input changes. Finding the derivative is key to identify where the function is increasing or decreasing.

For our function \( a(x) = \sqrt{-x + 4} \), we rewrite it as \( (-x + 4)^{1/2} \) for convenience in differentiation. Using derivative rules along with the chain rule, which is a method for differentiating compositions of functions, we find:

• \( a'(x) = \frac{-1}{2\sqrt{-x+4}} \)

This derivative informs us of the slope of the tangent to the curve at any given x-value within its domain. Understanding its sign will tell us whether \( a(x) \) is increasing or decreasing.

Chain Rule

The chain rule is a fundamental technique in calculus used when differentiating composite functions. Think of it as a way to "unpack" nested functions through differentiation.

Our function \( a(x) = (-x+4)^{1/2} \) demonstrates the use of the chain rule. Here, there's an inner function \(-x+4\) nested inside the outer square root function. The chain rule helps us differentiate by following these steps:

• Differentiate the outer function: For \((something)^{1/2}\), differentiate to \(\frac{1}{2}(something)^{-1/2}\).
• Differentiate the inner function with respect to x: \( -x+4 \) becomes \( -1 \).
• Multiply both results together: \( \frac{-1}{2\sqrt{-x+4}} \).

Thus, the chain rule simplifies the process of finding the derivative of composite functions.

Endpoints in Analysis

Endpoints are particularly important when analyzing function behavior, especially within a given domain. They are the boundary points where the behavior of the function can change.

In this exercise, the endpoint is \(x = 4\). It is vital to consider endpoints because they often represent where the domain starts or ends. In our function’s domain \((-\infty, 4]\), as we approach \(x = 4\), the derivative \( \frac{-1}{2\sqrt{-x+4}} \) becomes undefined because the denominator reaches zero.
However, this does not affect the overall behavior noted in the domain since the function decreases throughout and does not require continuity of the derivative at a single endpoint. Always watch how the function behaves as it nears endpoints because this can influence conclusions about increasing or decreasing intervals.