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Chapter 2: Problem 35

Show that \(f\) is continuous on the given interval. \(f(x)=\sqrt{x-4} ; \quad[4,8]\)

Short Answer

Expert verified
The function \( f(x) = \sqrt{x-4} \) is continuous on the interval \([4,8]\) because it is the composition of continuous functions over its domain.

Step by step solution

01

Understand the Problem

The task is to prove that the function \( f(x) = \sqrt{x-4} \) is continuous on the interval \([4, 8]\). A function is continuous on a closed interval if it is continuous at every point in that interval.
02

Check Domain of Function

The domain of the function \( f(x) = \sqrt{x-4} \) is determined by the condition under the square root. Since the square root is defined for non-negative numbers, we have \( x-4 \geq 0 \), or \( x \geq 4 \). Therefore, the domain of \( f \) is \([4, \infty)\). This includes our interval \([4, 8]\).
03

Use Properties of Continuous Functions

The function \( f(x) = \sqrt{x-4} \) is a composition of the function \( g(x) = x-4 \) and the square root function \( h(x) = \sqrt{x} \). Both of these functions are continuous where they are defined: \( g(x) \) is defined and continuous everywhere, and \( h(x) \) is defined and continuous for \( x \geq 0 \). Thus, their composition \( f(x) = h(g(x)) = \sqrt{x-4} \) is continuous where it is defined, including the interval \([4, 8]\).
04

Conclusion

Since all conditions for the square root function are met within the interval \([4, 8]\) and the composition is continuous, we conclude that \( f \) is continuous on the interval \([4, 8]\).

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Key Concepts

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

Square Root Function
A square root function involves the mathematical operation of finding the square root of a given expression. In our given problem, the function is expressed as \( f(x) = \sqrt{x-4} \). This means that for every value of \( x \), we subtract 4 and then find the square root of the result.

The square root function has specific characteristics:
  • It is only defined for non-negative values. For \( \sqrt{x-4} \) to be real and valid, \( x-4 \) must be greater than or equal to zero.
  • This leads to the requirement that \( x \geq 4 \) for our function.
  • As a result, this makes it a naturally restricted function.
  • Graphically, the square root function starts at a specific point and continues to rise gradually in a smooth curve.
Thus, understanding the square root function is crucial in defining where the function is meaningful and analyzing its behavior across intervals.
Function Domain
The domain of a function refers to all the possible input values (usually \( x \) values) for which the function is defined. Here, we consider the function \( f(x) = \sqrt{x-4} \).

Due to the nature of square roots:
  • We can only take the square root of numbers 0 or greater.
  • This sets a condition \( x-4 \geq 0 \), simplifying to \( x \geq 4 \).
  • Hence, the domain is \([4, \infty)\), meaning \( x \) can be any value from 4 up to infinity.
The domain is vital for understanding where the function can operate. Moreover, we must check that our interval of interest, \([4, 8]\), falls within this domain, ensuring the function behaves consistently over this range.
Composite Function
A composite function is created when one function is applied to the result of another function. In simpler terms, it's like a function inside a function. Our given function \( f(x) = \sqrt{x-4} \) is a composite function.

Here's how it breaks down:
  • The inner function \( g(x) = x-4 \) shifts the input by subtracting 4.
  • The outer function \( h(x) = \sqrt{x} \) then takes the square root of that shifted value.
  • So, \( f(x) = h(g(x)) = \sqrt{x-4} \).
  • Both \( g(x) \) and \( h(x) \) are continuous functions. Therefore, their composition is also continuous when defined.
This composition is particularly relevant for establishing continuity since each part of the function adheres to the conditions for being continuous on its domain.
Closed Interval
A closed interval, such as \([4, 8]\), is a range of values that includes the endpoints. In mathematical terms, it means that the interval contains all numbers between 4 and 8, as well as the numbers 4 and 8 themselves.

Here's what you need to know about closed intervals:
  • The notation \([a, b]\) indicates that both \( a \) and \( b \) are part of the interval.
  • For the function \( f(x) = \sqrt{x-4} \), we examine whether the function is continuous at every point in this closed interval.
  • Continuity on a closed interval ensures that the function has no breaks, jumps, or undefined points within the interval boundaries.
  • This is confirmed for \( f(x) = \sqrt{x-4} \) on \([4, 8]\) since it meets the necessary conditions discussed previously.
Understanding closed intervals helps in analyzing where exactly the function is continuous, providing insights into how the function behaves from the start to endpoint.

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Most popular questions from this chapter

Charles' law for gases states that if the pressure remains constant, then the relationship between the volume \(V\) that a gas occupies and its temperature \(T\) (in \(\left.{ }^{\circ} \mathrm{C}\right)\) is given by \(V=V_{0}\left(1+\frac{1}{273} T\right)\). The temperature \(T=-273^{\circ} \mathrm{C}\) is absolute zero. (a) Find \(\lim _{T \rightarrow-273^{*}} V\). (b) Why is a right-hand limit necessary?

A convex lens has focal length \(f\) centimeters. If an object is placed a distance \(p\) centimeters from the lens, then the distance \(q\) centimeters of the image from the lens is related to \(p\) and \(f\) by the lens equation $$\frac{1}{p}+\frac{1}{q}=\frac{1}{f}$$ As shown in the figure on the following page, \(p\) must be greater than \(f\) for the rays to converge. (a) Investigate \(\lim _{p \rightarrow f^{*}} q\). (b) What is happening to the image as \(p \rightarrow f^{+} ?\)

Exer. \(43-44:\) Show that \(f\) is continuous at the number \(a\). $$ f(x)=\sqrt{5 x+9} ; \quad a=8 $$

Find all numbers at which \(f\) is continuous. $$ f(x)=\frac{x}{\sqrt{1-x^{2}}} $$

Sketch the graph of \(f\) and find each limit, if it exists: (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 1^{+}} f(x)\) (c) \(\lim _{x \rightarrow 1} f(x)\) $$ f(x)=\left\\{\begin{array}{ll} -x^{2} & \text { if } x<1 \\ 2 & \text { if } x=1 \\ x-2 & \text { if } x>1 \end{array}\right. $$
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