Mister Exam
Graphing y = |3-√(2-x)|
The solution
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| _______| f(x) = |3 - \/ 2 - x |
$$f{\left(x \right)} = \left|{3 - \sqrt{2 - x}}\right|$$
f = Abs(3 - sqrt(2 - x))
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\left|{3 - \sqrt{2 - x}}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -7$$
so we need to solve the equation:
$$\left|{3 - \sqrt{2 - x}}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -7$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left|{3 - \sqrt{2 - x}}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{3 - \sqrt{2 - x}}\right| = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \left|{3 - \sqrt{2 - x}}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{3 - \sqrt{2 - x}}\right| = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(3 - sqrt(2 - x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{3 - \sqrt{2 - x}}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{3 - \sqrt{2 - x}}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left|{3 - \sqrt{2 - x}}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{3 - \sqrt{2 - x}}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\left|{3 - \sqrt{2 - x}}\right| = \left|{\sqrt{x + 2} - 3}\right|$$
- No
$$\left|{3 - \sqrt{2 - x}}\right| = - \left|{\sqrt{x + 2} - 3}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{3 - \sqrt{2 - x}}\right| = \left|{\sqrt{x + 2} - 3}\right|$$
- No
$$\left|{3 - \sqrt{2 - x}}\right| = - \left|{\sqrt{x + 2} - 3}\right|$$
- No
so, the function
not is
neither even, nor odd