Question

If f(x) = x^2 – 25 and g(x) = x – 5, what is the domain of (f/g)(x)

Answer 1

Answer:

D=(-∞,∞)

Step-by-step explanation:

We have the expressions:

$f(x)=x^2-25\\g(x)=x-5$

and we have to calculate $(\frac{f}{g})(x)$ to know the domain.

The domain of a function is all real numbers except where the expression is undefined.

Then,

$(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$

$\frac{f(x)}{g(x)}=\frac{x^2-25}{x-5}$

Observation: We can use difference of squares in f(x).

Difference of squares is: $a^2-b^2=(a+b)(a-b)$, then

$f(x)=x^2-25\\f(x)=x^2-5^2\\f(x)=(x+5)(x-5)$

Rewriting the equation:

$\frac{f(x)}{g(x)}=\frac{x^2-25}{x-5}\\\\=\frac{(x+5)(x-5)}{(x-5)}$

Simplifying:

$\frac{f(x)}{g(x)}=\frac{(x+5)(x-5)}{(x-5)}=x+5$

We can see that the expression is defined for all real numbers. Then the domain of $(\frac{f}{g})(x)$ is all real numbers.

Domain: D=(-∞,∞)

Answer 2

(f/g)(x) = (x^2-25)/(x-5) = x+5. So domain will be R, all real numbers