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Chapter 1: Problem 69

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

Short Answer

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

Step by step solution

01

Understand the function

The function given is \( a(x) = \sqrt{-x + 4} \). This is a square root function and the expression inside the square root, \(-x + 4\), must be non-negative for the function to be real-valued because the square root of a negative number is not defined in the set of real numbers.
02

Determine the domain

The domain of the function is obtained by setting \(-x + 4 \geq 0\). Solving \(-x + 4 = 0\) gives \(x \leq 4\), so the domain of the function is \((-\infty, 4]\).
03

Find the derivative

To determine where the function is increasing or decreasing, compute the derivative: \( a'(x) = \frac{d}{dx}\left(\sqrt{-x+4}\right)\). Use the chain rule: if \( u(x) = -x + 4 \), then the derivative \( a'(x) = \frac{1}{2\sqrt{u(x)}} \cdot (-1) \). So \( a'(x) = \frac{-1}{2\sqrt{-x+4}} \).
04

Analyze the sign of the derivative

The function is increasing where \( a'(x) > 0 \) and decreasing where \( a'(x) < 0 \). Since \( a'(x) = \frac{-1}{2\sqrt{-x+4}} \), the derivative is negative throughout its domain \((-\infty, 4)\) because the numerator is \(-1\) and the denominator \(2\sqrt{-x+4}\) is always positive for \(x \in (-\infty, 4)\).
05

Conclude the intervals of increase and decrease

Since \( a'(x) < 0 \) for all \( x \in (-\infty, 4) \), the function is decreasing throughout its domain. There are no intervals where the function is increasing.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain Determination
To determine the domain of a function, you need to figure out all the possible input values (x values) that will make the function work. For the function \( a(x) = \sqrt{-x + 4} \), we must ensure that the expression inside the square root, \(-x + 4\), is not negative.

This is because the square root of a negative number does not exist in the set of real numbers. To find when \(-x + 4 \geq 0\), solve the inequality:
  • Start by moving \( x \) to the other side: \(-x \geq -4\)
  • Multiply by \(-1\) and flip the inequality sign: \(x \leq 4\)
This tells us that the domain of the function is \((-\infty, 4]\). This means you can plug any number from negative infinity to 4 inclusive into the function, and it will yield a real value.
Derivative Analysis
Finding the derivative of a function helps us determine where the function is increasing or decreasing. The derivative essentially shows the rate at which the function's value is changing. For the function \( a(x) = \sqrt{-x + 4} \), we use the chain rule to differentiate it.

We set \( u(x) = -x + 4 \). The chain rule lets us say:
  • \( a'(x) = \frac{d}{dx} \left( \sqrt{u(x)} \right) \)
  • The derivative of \( \sqrt{u(x)} \) is \( \frac{1}{2 \sqrt{u(x)}} \times u'(x) \)
  • Since \( u(x) = -x + 4 \), \( u'(x) = -1 \)
Thus, \( a'(x) = \frac{-1}{2\sqrt{-x+4}} \).

Pay attention here 鈥� the value of the derivative and its sign will tell whether the original function is increasing or decreasing. A positive derivative indicates the function increases, while a negative one means it decreases.
Function Decreasing
The function's behavior over its domain can be identified by looking at the sign of the derivative. With the derivative \( a'(x) = \frac{-1}{2\sqrt{-x+4}} \), we notice a few key things:
  • The numerator \(-1\) is always negative.
  • The denominator \(2\sqrt{-x+4}\) is positive for all \(x\) in the domain \((-\infty, 4]\).
Consequently, the derivative \( a'(x) \) is negative for all \( x \leq 4 \). Since the derivative is negative throughout its domain, the function \( a(x) = \sqrt{-x + 4} \) is decreasing.

If you're asked whether the function is increasing anywhere, it's not! Some functions never increase; they just keep decreasing within their domain, as in this case. Always remember a negative sign in the derivative indicates a decline within the domain of concern.

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Most popular questions from this chapter

Let \(f(t)\) be the number of ducks in a lake \(t\) years after 1990 . Explain the meaning of each statement: a. \(f(5)=30\) b. \(f(10)=40\)

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