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Answer:
Domain:— [ x ≥ -2, x ≤ -3 ]
Range:— [ y ≥ 0 ]
Step-by-step explanation:
You may use graphing calculator to draw a graph and examine the graph’s domain and range. However, I’ll explain further about the graph of quadratic in a surd.
First, factor the quadratic expression in the surd:—
[tex]\displaystyle \large{f(x)=\sqrt{(x+3)(x+2)}}[/tex]
We can find the x-intercepts by letting f(x) = 0.
I’ll be separating in two parts — one for finding x-intercept and one for finding y-intercept.
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Finding x-intercepts
Let f(x) = 0.
[tex]\displaystyle \large{0=\sqrt{(x+3)(x+2)}}[/tex]
Solve for x, square both sides:—
[tex]\displaystyle \large{0^2 = (\sqrt{(x+3)(x+2)})^2}\\\displaystyle \large{0=(x+3)(x+2)}[/tex]
Simply solve a quadratic equation:—
[tex]\displaystyle \large{x=-3,-2}[/tex]
Therefore, x-intercepts are:—
[tex]\displaystyle \large{\boxed{x=-3,-2}}[/tex]
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Finding y-intercept
Let x = 0.
[tex]\displaystyle \large{f(x)=\sqrt{(0+3)(0+2)}}\\\displaystyle \large{f(x)=\sqrt{3 \cdot 2}}\\\displaystyle \large{f(x)=\sqrt{6}}[/tex]
Therefore, y-intercept is:—
[tex]\displaystyle \large{\boxed{\sqrt{6}}}[/tex]
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However, I want you to focus on x-intercepts instead. We know that the square root only gives you a positive value. That means the range of function can only be y ≥ 0.
For domain, first, we have to know how or what the graph looks like. You can input the function in a graphing calculator as you’ll see that when x ≥ -2, the graph heads to the right while/when x ≤ -3, the graph heads to the left. This means that the lesser value of x-intercept gets left and more value get right.
See, between -3 < x < -2, there is no curve, point or anything between the interval. Therefore, -3 < x < -2 does not exist in function.
Hence, the domain is:—
x ≥ -2, x ≤ -3
