a. The point (-5, 1) is not on the graph of f(x).
b. If x = 1, f(x) = 0.3846, and the corresponding point on the graph of f is (1, 0.3846).
c. The values of x where f(x) = 1 are x = √5 and x = -√5. The corresponding points on the graph of f are (√5, 1) and (-√5, 1).
d. The domain of f(x) is all real numbers.
e. The x-intercept of the graph of f is (0, 0).
f. The y-intercept of the graph of f is (0, 0).
(a) To check if the point (-5, 1) lies on the graph of f(x), we need to substitute x = -5 into the function and see if the resulting value equals 1.
f(-5) = (10)(-5)^2 / ((-5)^4 + 25) = 10(25) / (625 + 25) = 250 / 650 = 0.3846
Since f(-5) = 0.3846 and not 1, the point (-5, 1) is not on the graph of f.
(b) If x = 1, then f(x) = (10)(1)^2 / ((1)^4 + 25) = 10 / 26 = 0.3846
(c) To find the values of x where f(x) = 1, we set the function equal to 1 and solve for x:
10x^2 / (x^4 + 25) = 1
Multiplying both sides by (x^4 + 25) gives:
10x^2 = x^4 + 25
Rearranging the equation:
x^4 - 10x^2 + 25 = 0
Factoring the expression:
(x^2 - 5)^2 = 0
Therefore, x^2 - 5 = 0, which means x^2 = 5. Taking the square root of both sides, we get x = ±√5.
(d) The domain of a rational function is all real numbers except for the values of x that make the denominator zero. In this case, the denominator x^4 + 25 is never zero for any real number.
(e) The x-intercepts of a function are the points where the function crosses the x-axis, which means f(x) = 0. Setting f(x) to zero and solving for x:
10x^2 / (x^4 + 25) = 0
Since the denominator is never zero, the only way for the equation to be true is if the numerator is zero. Therefore, x^2 = 0, which means x = 0.
(f) The y-intercept of a function is the point where the function crosses the y-axis, which means x = 0. Setting x = 0 in the function:
f(0) = (10)(0)^2 / ((0)^4 + 25) = 0 / 25 = 0
Question:-
Answer the questions about the following function.
f left parenthesis x right parenthesis equals StartFraction 10 x squared Over x Superscript 4 Baseline plus 25 EndFractionf(x)=10x2 x4+25
(a) Is the point
left parenthesis negative StartRoot 5 EndRoot comma 1 right parenthesis−5,1
on the graph of f?
(b) If
x equals 1 commax=1,
what is f(x)? What point is on the graph of f?
(c) If
f left parenthesis x right parenthesis equals 1 commaf(x)=1,
what is x? What point(s) is (are) on the graph of f?
(d) What is the domain of f?
(e) List the x-intercepts, if any, of the graph of f.
(f) List the y-intercept, if there is one, of the graph of f.