Question

Let f(1, 3) `rightarrow` R be a function defined by f(x) = `(x[x])/(1 + x^2)`, where [x] denotes the greatest integer ≤ x, Then the range of f is ______.

Options

• `(2/5, 3/5) ∪ (3/4, 4/5)`

• `(2/5, 1/2) ∪ (3/5, 4/5)`

• `(2/5, 4/5)`

• `(3/5, 4/5)`

Answer 1

Let f(1, 3) `rightarrow` R be a function defined by f(x) = `(x[x])/(1 + x^2)`, where [x] denotes the greatest integer ≤ x, Then the range of f is `underlinebb((2/5, 1/2) ∪ (3/5, 4/5))`

Explanation:

Consider the given equation as,

f: (1, 3) `rightarrow` R

f(x) = `(x[x])/((x^2 + 1))`

Rewrite the above Equation as,

f(x) = `{{:(x/((x^2 + 1)), x ∈ (1, 2)),((2x)/((x^2 + 1)), x ∈ (2, 3)):}`  ...(1)

f'(x) = `{{:(((x^2 + 1) xx 1 - x xx 2x)/(x^2 + 1)^2, x ∈ (1, 2)),(((x^2 + 1) xx 2 - 2x xx 2x)/(x^2 + 1)^2, x ∈ (2, 3)):}`

f'(x) = `{{:((x^2 + 1 - x xx 2x)/(x^2 + 1)^2, x ∈ (1, 2)),((2 + 2x^2 - 4x^2)/(x^2 + 1)^2, x ∈ (2, 3)):}`

f'(x) = `{{:((1 - x^2)/(x^2 + 1)^2, x ∈ (1, 2)),((2(1 - x^2))/(x^2 + 1)^2, x ∈ (2, 3)):}`

From the above equation we get negative value . since, f(x) is decreasing function. substitute the value in the equation (1) we get the range of functions belonging to the,

f(x) = `x/((x^2 + 1)) x ∈ (1, 2)`

f(x) = `1/((1 + 1)) = 1/2`

f(x) = `2/((2^2 + 1)) = 2/5`

f(x) = `(2x)/((x^2 + 1)) x ∈ (2, 3)`

f(x) = `(2 xx 2)/((2^2 + 1)) = 4/5`

f(x) = `(2 xx 3)/((3^2 + 1)) = 6/10 = 3/5`

Hence R_f = `∈ (2/5, 1/2) ∪ (3/5, 4/5)`