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Question
Find the domain and range of the following function.
f(x) = 7x2 + 4x − 1
Solution 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, ∞)`
Solution 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, ∞).`
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