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Chapter 3: Problem 42

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=\sqrt{4-x} $$

Short Answer

Expert verified
Domain is \((-\infty, 4]\); Range is \([0, \infty)\); \(f(x)\) decreases over its domain.

Step by step solution

01

Understanding the Function

The function given is \( f(x) = \sqrt{4-x} \). It is a square root function combined with a linear term \( -x \). Square root functions are only defined for non-negative values under the square root.
02

Find the Domain of the Function

To ensure the expression under the square root is non-negative, set the inequality: \[ 4-x \geq 0 \] Solve for \( x \):\[ x \leq 4 \] Thus, the domain \( D \) is \( (-\infty, 4] \).
03

Find the Range of the Function

Since \( f(x) = \sqrt{4-x} \) only takes real non-negative values, we start with the maximum of \( x \), which is 4 (the endpoint), resulting in \( f(4) = 0 \). As \( x \) decreases from 4 to negative infinity, the \( f(x) \) increases without bound. Hence, the range \( R \) of \( f \) is \( [0, \infty) \).
04

Sketch the Graph of the Function

To sketch \( f(x) = \sqrt{4-x} \), note when \( x = 4 \), \( f(x) = 0 \). The graph starts at \((4,0)\) and rises to the left since the square root increases as \( x \) decreases from 4. The graph is a half-parabola opening downwards.
05

Determine the Intervals of Increase, Decrease, or Constancy

Calculate the derivative: \[ f'(x) = \frac{d}{dx}[\sqrt{4-x}] = -\frac{1}{2\sqrt{4-x}} \]Since \( f'(x) < 0 \) for all \( x < 4 \), this means the function \( f(x) \) is always decreasing on its entire domain \((-\infty, 4]\).

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Key Concepts

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

Domain and Range
When dealing with square root functions like \( f(x) = \sqrt{4-x} \), it's crucial to understand the concepts of domain and range. The **domain** of a function represents all possible input values (\( x \)-values) that the function can accept without leading to mathematical inconsistencies.
For square root functions, we must ensure that the value under the square root sign is non-negative because the square root of a negative number isn't defined in the set of real numbers.
  • To find the domain of \( f(x) = \sqrt{4-x} \), we solve the inequality \( 4-x \geq 0 \), which simplifies to \( x \leq 4 \).
  • This means the domain of \( f \) is \( (-\infty, 4] \).
The **range** of a function is all possible output values (\( f(x) \)-values). For \( f(x) = \sqrt{4-x} \), note that:
  • The square root function outputs only non-negative numbers, starting at 0 when \( x = 4 \).
  • Therefore, the range of \( f \) starts from 0 and extends to positive infinity: \( [0, \infty) \).
Graph Sketching
Sketching the graph of a square root function helps visually understand its behavior. For the function \( f(x) = \sqrt{4-x} \), start by identifying key points and the general shape.
  • At \( x = 4 \), the calculation \( f(4) = \sqrt{4-4} = 0 \) gives the point \( (4, 0) \).
  • Since the function includes a negative linear term \(-x\), the graph's general motion is upwards as \( x \) decreases from 4.
The shape of this graph will resemble a half-parabola, opening downwards as it moves left from the point \( (4, 0) \). Keep in mind:
  • The graph never dips below the \( x \)-axis because square roots yield only non-negative outputs.
  • As \( x \) approaches negative infinity, \( f(x) \) continues to rise.
Interval Analysis
Interval analysis involves inspecting the behavior of functions over specific sections of the \( x \)-axis.
By differentiating \( f(x) = \sqrt{4-x} \), you can determine whether the function is increasing, decreasing, or constant along its domain.The derivative \( f'(x) = -\frac{1}{2\sqrt{4-x}} \) informs us about the rate of change:
  • Because \( f'(x) < 0 \) for all \( x < 4 \), it is clear that \( f(x) \) is always decreasing in the domain \((-\infty, 4]\).
  • At no point does the derivative equal zero or become positive, which means the function does not have any turning points or increasing intervals.
In summary, \( f(x) \) keeps descending throughout its domain, and this information is valuable when predicting the behavior of the function for both small-scale and larger patterns. Understanding this behavior can guide in making estimations and informed conclusions about the outputs given different inputs in the function's range.

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