Match the following lists and then choose the correct code.

Options

A. `{:(a,b,c,d),(p,r,s,q):}`
B. `{:(a,b,c,d),(r,s,q,p):}`
C. `{:(a,b,c,d),(r,q,s,p):}`
D. `{:(a,b,c,d),(p,q,s,r):}`

Answer

B

Explanation

a. `f(x)=log_(3)(5+4x-x^(2))`
`=log_(3)(9-(x-2)^(2))`
Now, `-oo lt 9 -(x-2)^(2) le 9`
But for `f(x)` to get defined, `0 lt 9-(x-2)^(2) le 9`
`or -oo lt log_(3)(9-(x-2)^(2)) le log_(3) 9`
`or -oo lt log_(3)(9-(x-2)^(2)) le 2`
Hence, the range is `(-oo, 2].`
b. `f(x)=log_(3)(x^(2)-4x-5)`
`=log((x-2)^(2)-9)`
For `f(x)` to get defined, `o lt (x-2)^(2)-9 lt oo`
or `underset (x to 0)(lim)logx lt log_(e)(x-2)^(2)-9 lt underset(x to oo)(lim) log x`
`or -oo lt f(x) lt oo`
Hence, the range is R.
c. `f(x)=log_(3)(x^(2)-4x+5)`
`=log_(3)((x-2)^(2)+1)`
`(x-2)^(2)+1 in [1,oo)`
or `log_(3)(x^(2)-4x+5) in [0,oo)`
d. `f(x) =log_(3)(4x-5-x^(2))`
`=log_(3)(-5-(x^(2)-4x))`
`=log_(3)(-1-(x-2)^(2))`
Now, `-1-(x-2)^(2) lt 0` for all x.
Hence, the function is not defined.