Question

Given f of x is equal to the quantity x plus 3 end quantity over the quantity x squared plus 2 times x minus 3 end quantity and g(x) = log4x, evaluate (g – f )(2).

Answer 1

To evaluate (g - f)(2), we first need to find f(2) and g(2).

We start by finding f(2):

f(x) = (x + 3) / (x^2 + 2x - 3)

f(2) = (2 + 3) / (2^2 + 2(2) - 3) = 5 / 7

Next, we find g(2):

g(x) = log4x

g(2) = log4(2)

Now we can evaluate (g - f)(2):

(g - f)(2) = g(2) - f(2) = log4(2) - 5/7

Using a calculator, we can approximate log4(2) to be approximately -0.415.

So, (g - f)(2) is approximately:

(g - f)(2) ≈ -0.415 - 5/7 ≈ -1.125

Therefore, (g - f)(2) is approximately -1.125.

Answer 2

-
`(1)/(2)`

Explanation:

(g - f)(2)

= g(2) - f(2)

to find g(2) use the rule of logarithms

`log_(b)` x = n ⇒ x =
`b^(n)`

then

`log_(4)` 2 = n

2 =
`4^(n)`

2 =
`(2^2)^(n)`

2 =
`2^(2n)`, so equating exponents gives

2n = 1 ( divide both sides by 2 )

n =
`(1)/(2)`

so g(2) =
`(1)/(2)`

and

f(2) =
`(2+3)/(2^2+2(2)-3)` =
`(5)/(4+4-3)` =
`(5)/(5)` = 1

Then

(g - f)(2)

= g(2) - f(2)

=
`(1)/(2)` - 1

= -
`(1)/(2)`