Find the value of a for which function `f(x) = (ax^(2) + 3x - 4)/(3x - x^(2) +a)` has range off real numbers.

Options

A. `a le 1 or a ge 7`
B. `a ge 1 or a le 7`
C. `1 le a le 7`
D. none of these

Answer

C

Explanation

Let `y=(ax^(2)+3x -4)/(a + 3x -4x^(2))` Then, y takes all real values for ral values of x.
Now, `y=(ax^(2)+3x-4)/(a+3x-4x^(2))`
`rArr" "x^(2)(a+4y)+3x(1-y)-(4+ay) = 0` for all y `in` R
`rArr" "9(1-y)^(2) -4(a+4y)(4 + ay) ge 0` for all y `in R" "[because x in R therefore "Disc" ge 0]`
`rArr" "(9+16a)y^(2) + (4a^(2)+46)y + (9 + 16a) ge 0` for all y `in` R
`rArr" "9+16a gt 0 and (4a^(2)+46)^(2) - 4(9 + 16a)^(2) le 0`
`rArr" "9 + 16a gt 0 and 4(a+4)^(2) (a^(2) - 8a + 7) le 0`
`rArr" "a ge -(9)/(16) and a^(2) - 8a + 7 le 0`
`rArr" "a ge -(9)/(16) and le a le 7 rArr 1 le a le 7`