Question

Find the domain and range of the following function.

f(x) = 7x² + 4x − 1

Answer 1

f(x) = 7x² + 4x − 1

f is defined for all x

∴ Domain of f = R (i.e. the set of real numbers)

7x² + 4x − 1

= `7(x^2 + 4/7x) - 1`

= `7(x + 2/7)^2 - 1 - 4/7`

= `7(x + 2/7)^2 - 11/7 ≥ - 11/7`

∴ Range of f = `[-11/7, ∞)`

Answer 2

f(x) = 7x² + 4x - 1

f(x) is defined for all x ∈ R

∴ Domain = R

7x² + 4x − 1

`= 7[x^2 + 4/7x] - 1`

`= 7[x^2 + (4x)/7 + 4/49] - 28/49 - 1`

`= 7(x + 2/7)^2 - 77/49`

`= 7(x + 2/7)^2 - 11/7`

∴  `7(x + 2/7)^2 ≥ 0  "for all  x ∈ R"`

∴  `7(x + 2/7)^2 - 11/7 ≥ 0 - 11/7 "for all  x ∈ R"`

∴ `f(x) ≥ - 11/7 "for all  x ∈ R"`

∴ Range = `[-11/7, ∞)`

∴ Domain = R, Range = `[-11/7, ∞).`