What are the domain and range of the function f(x)=-log(5-x)+9

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