Question

Write the range of each of the following functions in interval notation

a) f(x) = x^2-7

b) g(x) =(√x)+5

c) h(x) =3x^2+1

Answer 1

Range : The set of values of the dependent variable for which a function is defined.

For a parabola ax^2 + bx + c = 0 with vertex (x, y)

If a < 0 then the range is f(x) <= y

If a > 0 then the range is f(x) >= y

a)

f(x)  =  x^2 - 7

Substitute x = 0 in f(x)

f(0)  =  0^2 - 7

y  =  - 7

The vertex is (0, -7) and a = 1 > 0

Hence, 

Range : f(x)  >=  - 7 (or)  [- 7, infnity )

b)

g(x) =(√x)+5

The range of radical function of the form c√(ax + b)  +  k is f(x)  >=  k

Here k = 5

Hence, 

Range : g(x) > =  5  (or)  [ 5, infinity )

c)

h(x)  =  3x^2 + 1

Substitute x = 0 in f(x)

h(0)  =  3(0^2) + 1

y  =  1

The vertex is (0, 1) and a = 3 > 0

Hence, 

Range : h(x)  >=  1 (or)  [1, infnity )

Answer :

a)  Range : f(x)  >=  - 7 (or)  [- 7, infnity )

b) Range : g(x) > =  5  (or)  [ 5, infinity )

c)  Range : h(x)  >=  1 (or)  [1, infnity ).