answer the questions about the following function

Question

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.

Answer 1

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