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calculus

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answer the questions about the following function
f(x)= 10x^2/x^4+25

a. is the point (-sqrt 5,1) on the graph
b. if x=3, what is f(x)? what point is on the graph of f?
c. if f(x)=1, what is x? what points are on the graph?
d. what is the domain of f?
e. list the x-intersepts? if any on the graph of f.
f. list the y-inntercept, if any on the graph of f.

Please help me understand this, and show work please.

  • calculus -

    a.

    When x = sqrt ( 5 )

    f ( x ) = 10 x ^ 2 / ( x ^ 4 + 25 ) =

    10 * sqrt ( 5 ) ^ 2 / ( sqrt ( 5 ) ^ 4 + 25 ) =

    10 * 5 / ( 25 + 25 ) = 50 / 50 = 1

    yes


    b.

    When x = 3

    f ( x ) = 10 x ^ 2 / ( x ^ 4 + 25 ) =

    10 x ^ 2 / ( x ^ 4 + 25 ) =

    10 * 3 ^ 2 ( 3 ^ 4 + 25 ) =

    10 * 9 / ( 81 + 25 ) =

    90 / 106 = 2 * 45 / ( 2 * 53 ) = 45 / 53


    c.

    10 x ^ 2 / ( x ^ 4 + 25 ) = 1 Multiply both sides by ( x ^ 4 + 25 )

    10 x ^ 2 = x ^ 4 + 25

    Substitution:

    x ^ 2 = z

    x ^ 4 = z ^ 2

    10 z = z ^ 2 + 25

    Exact solutions are :

    z = + OR - 5

    x ^ 2 = z

    x = + OR - sqrt ( 5 )

    d.

    x ^ 4 + 25 differently of = 0

    x ^ 4 are olway positive and

    x ^ 4 + 25 are olvay positive so Domain = all real numbers


    e.

    x-intersept = point where f ( x ) = 0

    In this case 10 x ^ 2 = 0

    Obviously x = 0 so

    x -inntercept point ( 0 , 0 )

    f.

    y-inntercept = point where x = 0

    In this case :

    f ( x ) = 10 * 0 ^ 2 / ( 0 ^ 2 + 25 ) =

    10 * 0 / ( 0 + 25 ) = 0 / 25 = 0

    y-inntercept point also ( 0 , 0 )



    P.S:

    If don't know how to solve equation:

    10 z = z ^ 2 + 25

    In google type:
    quadratic equation online

    When you see list of results click on:

    Free Online Quadratic Equation Solver:Solve by Quadratic Formula

    When page be open in rectangle type:

    10 z = z ^ 2 + 25

    and click option solve it

    You will see solution step-by-step

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