Function: f(x) = – log(5 – x) + 9
Restriction for the domain of f:
5 – x > 0
x < 5
So the domain of f is
Dom(f) = {x ∈ R: x < 5}
or using the interval notation,
Dom(f) = ]– ∞, 5[.
________
Next, we want to show that for any given y ∈ R, there is always an x ∈ Dom(f) so that
y = f(x)
Once we show that, we can conclude that the range of f is R (all real numbers).
Solving the equation for x:
y = f(x)
y = – log(5 – x) + 9
y – 9 = – log(5 – x)
– (y – 9) = log(5 – x)
9 – y = log(5 – x)
5 – x = 10^(9 – y)
x = 5 – 10^(9 – y)
Since there is not any restriction for y, the range of f is R (all real numbers).
I hope this helps. =)
Tags: logarithmic composite function domain range inverse solve algebra
