Chapter 1.3, Problem 5E

To determine

The points of discontinuity of the function. Identify the type of

Discontinuity.

Answer to Problem 5E

The function is continuous everywhere and discontinuity at $\left(-\infty ,-\frac{3}{2}\right)$ .

Explanation of Solution

Given:

  $y=\sqrt{2x+3}$

Calculation:

The given equation $y=\sqrt{2x+3}$ is a composition of two functions, $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=2x+3$ . Both, the functions are continuous. So, the function is continuous.

Find the discontinuity point of the function$y=\sqrt{2x+3}$:

Set the radicand in $\sqrt{2x+3}$ less than $0$ to find where the expression is undefined.

  $2x+3<0$

Solve for $x$ ,

  $x<\frac{-3}{2}$

The equation is undefined where the denominator equals $0$ , the argument of a square

root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

  $x<\frac{-3}{2}$

Thus, the function continuous everywhere and discontinuity at $\left(-\infty ,-\frac{3}{2}\right)$ .