How would I find the domain and range

Question

Update : f(x) = 5/2x^2 + 5x - 3

and

f(x) = -3x^2 + 5  

Answer 1

The rational function

We know that all possible values of x  is domain of a function.

To find which number make the fraction undefined create an equation where the denominator is not equal to zero.

Answer 2

Contd...

To find the range, we want to find all yy  for which there exists an x such that

We can solve this equation for x .

a = 2y , b = 5y, c = -3y - 5.

Roots are

So,for a given y, y  is in the range if this expression yields a real number.That is if

Range of

.

Answer 3

Contd...

• The quadratic function f (x) = - 3x ²  + 5

y = - 3x ²  + 5

The above function represents a parabola in standard form y  = ax ²+ bx  + c .

 a = - 3 , b = 0 and c  = 5.

Axis of symmetry x = -b/2a = 0/2(-3) = 0

Substitute x  = 0 in y  = - 3x ²  + 5.

y  = - 3(0)²  + 5

y  = 5

Vertex (x, y )= (0, 5)

a  is negative number the parabola opens down and has maximum value.

When the parabola opens up it has a maximum point which is the vertex of parabola (0, 5)

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

 parabola domain x  =  all real numbers.

In the maximum point y  = 5  so the graph of parabola cannot be upper than 5.

Thus the range of function y  ≤ 5.

Domain of function is all real numbers.

Range of the function is  {y |y  ≤ 5}.