A function f is such that f (x) = ln (5×-10), for x >2.
(i) State the range of f (ii) Find f^-1 (x)
(iii) State the range of f^-1 (iv) Solve f (x) = 0
Ok so i managed to do (ii) and (iv). However I couldn't do (i) and (iii). They say x >2 so i sub x equal to 2 in (i) and (iii) and I couldn't get the answer. The correct answer - (i) f>0 (iii) f (x)^-1 >2. My answer - (i) undefined (iii) f ^-1 (x) > 3.47781122. Please help. Thank you so much
(1)When you are given a domain for f(x), the x>2 in your case, you may or may not be able to put in the value of 2. In this case you can't since you would get ln(0) which is undefined. But suppose we had x equal to 2 plus a small amount, say x=2+d/5, where d is positive. In this case 5x-10 = d and we would have ln(d). If d were very small (but still positive), x would be greater than 2 and f(x) would be negative [it would actullay become very large negative a d got closer and closer to zero, i.e. f(x) -> -infinity as x->2+. That gives us the lower limit of (x) and the lower limit of its range. So it looks like the answer you gave for (i) is incorrect since, for example, when x=5.001, f(x) is negative. To find the upper bound on f(x), imagine x becoming very very large (the other end of x>2). What happens to f(x)? As an aside, we note that the reason for the domain of x being x>2 is that 5x-10, the argument of the log function, must be positive.
(ii) Generally what will work on a problem for finding the inverse of a function is to swap x and 'y' [=f(x)] and rearrange the formula. [Sometimes it gets so complicated, it doesn't work but it will work here.] So we have f(x) = y = ln(5x-10) To find the inverse, swap x and y and write x=ln(5y-10). You should know that if x=ln(5y-10), then e
x = 5y-10. Rearrange this to get y=\(\displaystyle f^{-1}(x)=\frac{1}{5}\, e^x\) + 2. There does't appear to be any restriction on x so what happens when x becomes very large negative? What about very large positive. Note: Typically the way to do this is just to interchange the range and domain of f(x). The way I have here may extend the range [it doesn't this time but it could]. So be careful about what the the question wants as an answer.
(iii) See (ii)
(iv) Either one of two ways: Use the inverse function or find the x which makes ln(5x-10)=0.
Good Luck
