find the domain and range of a function 9x^2+3x+5

Question

it is the question of mathematics.

Answer 1

Given function is f(x) = y = 9x^2+3x+5

For x = -3

y = 9(-3)^2+3*-3+5

y = 81-9+5= 77

For x = -2

y = 9(-2)^2+3(-2)+5

y = 36-6+5 = 35

For x = -1

y = 9(-1)^2+3(-1)+5

y = 9-3+5 = 11

For x = 0

y = 9(0)^2+3*0+5 = 0+0+5 = 5

For x = 1

y = 9(1)^2+3(1)+5

y = 9+3+5 = 17

For x = 2

y = 9(2)^2+3*2+5

y = 36+6+5 = 47

For x = 3

y = 9(3)^2+3(3)+5

y = 81+9+5

y = 95

x value is half coordinate pair and y value is half coordinate pair.

If x = -3,-2,-1,0,1,2,3 respectively substitute the x values in given function the set of coordinate pair is

be 77,35,11,5,17,47,95.

We know that x values is domain of the function and y values is range of the function.

Domain set is {-3,-2,-1,0,1,2,3}

Range set is {77,31,11,5,17,47,95}

Answer 2

The function y = f ( x) = 9x ^2 + 3x + 5.

The function f ( x) = 9x ^2 + 3x + 5 is a quadratic function or parabola function.There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

The domain of 9x ^2 + 3x + 5 is all real numbers.

• We first put the equation in to the form for a translated parabola y = a (x - h )^2 + k .

Center (h, k ).

In the next step we factored 9 from the right hand side to make the coefficient of our x  "+1" as this is the standard form.

y  = 9(x ^2 + x /3 + 5/9)

 To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression

 Here x coefficient = 12. so, (half the x coefficient)² = ((1/3)/2)²= 1/36.

Add and subtract 1/36 to x ^2 + x /3 + 5/9.

y  = 9(x ^2 + x /3 + 1/36 + 5/9 - 1/36)

y  = 9[ (x + 1/6)^2 + 5/9 - 1/36 ]

y  = 9(x + 1/6)^2 + 19/4

The above function represents a parabola vertex form  y = a (x - h )^2 + k .

 a  = 9 , h  = - 1/6 and k  = 19/4.

a  is positive number the parabola opens up and has minimum value.

When the parabola opens up it has a minimum point which is the vertex of parabola (- 1/6, 19/4)

In the minimum point y  = 19/4,  so the graph of parabola cannot be lower than 19/4.

Thus the range of function y  ≥ 19/4.

Range of the function is  {y |y  ≥ 19/4}.