Answer:
Step-by-step explanation:
(a) The function ...
[tex]f(x)=\dfrac{16x^{2}}{x^{4}+64}[/tex]
can be evaluated for x=-2√2 to get ...
[tex]\displaystylef(-2\sqrt{2})=\frac{16(-2\sqrt{2})^{2}}{(-2\sqrt{2})^4+64}=\frac{128}{64+64}=1[/tex]
The point (-2√2, 1) is on the graph of f(x).
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(b) Likewise, we can evaluate for x=2:
[tex]f(2)=\dfrac{16\cdot 2^2}{2^4+64}=\dfrac{64}{80}=0.8[/tex]
The point on the graph is (2, 0.8).
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(c) From part (a), we know that f(-2√2) = 1. Since the function is even, this means that f(2√2) = 1. The graph is a maximum at those points, so there are no other values for which f(x) = 1.
The points (±2√2, 1) are on the graph.
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(d) There are no values of x for which f(x) is undefined. The domain is all real numbers.
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(e) The only x-intercept is at the origin, (0, 0). The x-axis is a horizontal asymptote.
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(f) The only y-intercept is at the origin, (0, 0).
At [tex]x = -2\sqrt{2}[/tex] the above function f(x) = 1, at x = 2 the given function f(x) = 0.8, the domain of f are all real numbers, x-intercept and y-intercept is at the origin (0,0).
Given :
[tex]\rm f(x) = \dfrac{16x^2}{x^4+64}[/tex]
a) At [tex]x = -2\sqrt{2}[/tex] the above function becomes.
[tex]\rm f(-2\sqrt{2} ) = \dfrac{16(-2\sqrt{2})^2}{(-2\sqrt{2})^4+64}[/tex]
Simplify the above expression in order to determine the value of [tex]\rm f(-2\sqrt{2})[/tex].
[tex]\rm f(-2\sqrt{2})=\dfrac{16\times 8}{64+64}=\dfrac{128}{128}=1[/tex]
b) At x = 2 the given function becomes.
[tex]\rm f(2) = \dfrac{16\times (2)^2}{2^4+64}=\dfrac{64}{80}=0.8[/tex]
c) f(x) is maximum when at x = [tex]\pm 2 \sqrt{2}[/tex] and at this point f(x) = 1.
So, the point ([tex]\pm2\sqrt{2}[/tex] , 1) is on the graph of f.
d) f is defined for all values that are the domain of f is all real numbers.
e) Substitute f(x) = 0 in the given function.
[tex]\rm 0 = \dfrac{16x^2}{x^4+64}[/tex]
By simplifying the above equation, the value of x is zero.
So, the x-intercept is at the origin (0,0).
f) The y-intercept is also at origin (0,0).
For more information, refer to the link given below:
https://brainly.com/question/16875632
