Find all values of x satisfying the given conditions.
-$$y = \sqrt { 2 x + 3 } - \sqrt { x + 1 } - 1$$
A) No x-intercepts
B) 3
C) $- 3 , - 1$
D) $3 , - 1$

Question

Quizplus · Accepted Answer

The expression under the first radical must be non-negative, so $2x + 3 \ge 0$, which gives $x \ge -3/2$. The expression under the second radical must also be non-negative, so $x + 1 \ge 0$, which gives $x \ge -1$. Therefore, the domain of the function is $[-3/2, \infty)$.To find the x-intercepts, we set $y = 0$ and solve for $x$:$$0 = \sqrt { 2 x + 3 } - \sqrt { x + 1 } - 1$$$$\sqrt { 2 x + 3 } = \sqrt { x + 1 } + 1$$$$2 x + 3 = x + 1 + 2\sqrt { x + 1 } + 1$$$$x + 2 = 2\sqrt { x + 1 }$$$$(x + 2)^2 = 4(x + 1)$$$$x^2 + 4x + 4 = 4x + 4$$$$x^2 = 0$$$$x = 0$$Therefore, the only x-intercept is $x = 0$, which is not in the domain of the function. Therefore, the function has no x-intercepts.

Find All Values of X Satisfying the Given Conditions y=2x+3−x+1−1y = \sqrt { 2 x + 3 } - \sqrt { x + 1 } - 1y=2x+3​−x+1​−1

Find All Values of X Satisfying the Given Conditions $y = \sqrt { 2 x + 3 } - \sqrt { x + 1 } - 1$