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Chapter 2: Problem 25

At what points are the functions continuous? $$y=\sqrt{2 x+3}$$

Short Answer

Expert verified
The function is continuous for \(x \geq -\frac{3}{2}\).

Step by step solution

01

Understand the Function's Domain

The function given is \( y = \sqrt{2x + 3} \). We know that a square root function is defined only for non-negative numbers. Therefore, the expression inside the square root, \(2x + 3\), must be non-negative.
02

Determine the Domain of the Function

To find when \(2x + 3 \geq 0\), solve for \(x\):\[ 2x + 3 \geq 0 \] \[ 2x \geq -3 \] \[ x \geq -\frac{3}{2} \] Thus, the domain of the function is \(x \geq -\frac{3}{2}\).
03

Define Continuity of the Function

A function is continuous at every point in its domain. Since square root functions are continuous wherever they are defined, \(y = \sqrt{2x + 3}\) is continuous for all values of \(x\) in its domain \([ -\frac{3}{2}, \infty)\).

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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, like the one given by the formula \(y = \sqrt{2x + 3}\), involves taking the square root of an expression. The essential characteristic of a square root function is that it is only defined for non-negative numbers, since square roots of negative numbers are not real in the context of real-valued functions.
This means that whatever is inside the square root, known as the radicand, must be zero or positive to have a real number as a result.
  • When solving problems involving square roots, always start by ensuring the radicand is non-negative. This will safeguard against undefined values in real numbers.
  • Square root functions naturally have a domain that restricts input values to only those that make the radicand non-negative.
Understanding square root functions also means recognizing their continuity within their domain, meaning they are smooth and unbroken over all values that satisfy the non-negativity condition.
Function Domain
The domain of a function refers to all the possible input values \(x\) that will produce a real number output when substituted into the function. For the square root function \(y = \sqrt{2x + 3}\), determining the domain is critical to find where the function is defined and hence, continuous.
  • The domain is found by setting the inside of the square root expression \(2x + 3\) greater than or equal to zero, since the square root of a negative value isn't defined in real numbers.
  • In this exercise, solving \(2x + 3 \geq 0\) gives \(x \geq -\frac{3}{2}\).
This means the domain of this function 鈥� the set of all \(x\) values for which the function gives a real number 鈥� is \([-\frac{3}{2}, \infty)\). This indicates the function is defined and continuous for all \(x\) values starting from \(x = -\frac{3}{2}\) and extending to positive infinity.
Non-negative Numbers
Non-negative numbers are key in determining when certain functions, such as the square root, are defined. Non-negative numbers include all positive numbers and zero, excluding any negative values.
This is essential because certain mathematical operations, like square roots, require non-negative inputs to be defined in the real number system.
  • In this context, the expression \(2x + 3\) inside the square root must result in a non-negative number.
  • Mathematically, this requires solving inequalities to ensure the value is at least zero, a process often used to define domains for functions involving squares or roots.
By focusing on non-negative values, we ensure that every evaluation of the function like \(\sqrt{2x + 3}\) yields a real number, smoothly mapping the input domain to the output range without interruption or undefined results.

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Most popular questions from this chapter

Use the Intermediate Value Theorem to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$\sqrt{x}+\sqrt{1+x}=4$$

Suppose that \(f(x)\) and \(g(x)\) are polynomials in \(x\) and that \(\lim _{x \rightarrow \infty}(f(x) / g(x))=2 .\) Can you conclude anything about\(\lim _{r \rightarrow \infty}(f(x) / g(x)) ?\) Give reasons for your answer.

Domains and Asymptotes Determine the domain of each function. Then use various limits to find the asymptotes. $$y=\frac{x^{3}}{x^{3}-8}$$

Find the limits. Write \(\infty\) or \(-\infty\) where appropriate. \(\lim \frac{x^{2}-3 x+2}{x^{3}-4 x}\) as a. \(x \rightarrow 2^{+}\) b. \(x \rightarrow-2^{+}\) c. \(x \rightarrow 0^{-}\) d. \(x \rightarrow 1^{+}\) e. What, if anything, can be said about the limit as \(x \rightarrow 0 ?\)

Domains and Asymptotes Determine the domain of each function. Then use various limits to find the asymptotes. $$y=\frac{4 e^{x}+e^{2 x}}{e^{x}+e^{2 x}}$$
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