Question

Determine the domain and range of the function .

Answer 1

The function is f(x) = (x² + 15x - 56)/(x² + 3x + 2).

Domain :

Domain of a function f(x) is set of those values of x which will make the function mathematically legal or correct..certain operations like division by zero , square root of a negative number do not exist in real maths.

1. Domain excludes x - values that result in division by zero.

2. Domain excludes x - values that result in even roots of negative numbers.

Find the factors of numerator expression x² + 15x - 56.

The above expression cannot be write in factorization form, since the any two numbers does not adjusted in their sum 15 and their product - 56.

Find the factors of denominator expression x² + 3x + 2.

= x² + 3x + 2

= x² + 2x + x + 2

= x(x + 2) + 1(x + 2)

= (x + 1)(x + 2).

f(x) = (x² + 15x - 56) / [(x + 1)(x + 2)].

The denominator expression is (x + 1)(x + 2) and it is equals to zero.

(x + 1)(x + 2) = 0

x = - 1 and x = - 2.

The denominator is zero, when x = - 1 and x = - 2.

Domain excludes x - values that result in division by zero.

The function has an implied domain that consists of all real x other than x = - 1 and x = - 2.These values are excluded from the domain because division by zero is undefined.

Domain : {x Є R : x ≠ - 2 and x ≠ - 1}.

Answer 2

To determine the range of f (x ) =

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

• Step-1

y = 0 in our range

To find x intercept let the numarator = 0

Compare it to

Answer 3

Contiuous...

• Step-2

We can solve this equation for x

If y not equals to 0, this is a quadratic equation in x ,so we can sole it with the quadratic formula:

Compare it to

a = y-1 , b = 3y - 15, c = 2y + 56.

Roots are

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

We exclude y = 0 earlier , but we know y = 0 in our range

Thus the range of f(x)=  

.