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Given the function f left parenthesis x right parenthesis equals StartRoot x minus 10 EndRoot

​,
​(a) Find f Superscript negative 1 Baseline left parenthesis x right parenthesis
.
​(b) Graph f and f Superscript negative 1

in the same rectangular coordinate system.
​(c) Use interval notation to give the domain and the range of f and f Superscript negative 1
.
​(Hint: To solve for a variable involving an nth​ root, raise both sides of the equation to the nth​ power, left parenthesis RootIndex n StartRoot y EndRoot right parenthesis Superscript n Baseline equals y
​.)

User Lemmy
by
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1 Answer

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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.

User Primer
by
8.7k points
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