Get detailed and reliable answers to your questions on IDNLearn.com. Our community is here to provide the comprehensive and accurate answers you need to make informed decisions.

Write the domain of the function in interval notation. Write numbers as integers or simplified fractions.

[tex] f(x)=\frac{x^2-16}{x-4} [/tex]

The domain of [tex] f(x) [/tex] is [tex] \square [/tex].


Sagot :

To find the domain of the function [tex]\( f(x) = \frac{x^2 - 16}{x - 4} \)[/tex], we should first look for any values of [tex]\( x \)[/tex] that would make the denominator zero, as these would cause the function to be undefined.

1. Identify the points where the function is undefined:
The denominator of our function is [tex]\( x - 4 \)[/tex]. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 4 = 0 \implies x = 4 \][/tex]
So, the function is undefined at [tex]\( x = 4 \)[/tex].

2. Simplify the function if possible:
Next, simplify the numerator [tex]\( x^2 - 16 \)[/tex]. Notice that the numerator can be factored using the difference of squares:
[tex]\[ x^2 - 16 = (x - 4)(x + 4) \][/tex]
So, our function can be rewritten as:
[tex]\[ f(x) = \frac{(x - 4)(x + 4)}{x - 4} \][/tex]
For [tex]\( x \neq 4 \)[/tex], the [tex]\( x - 4 \)[/tex] terms cancel out, leaving:
[tex]\[ f(x) = x + 4 \text{ for } x \neq 4 \][/tex]
Although we have simplified the function, we must remember that the original function was undefined at [tex]\( x = 4 \)[/tex].

3. Write the domain in interval notation:
Since the function is undefined at [tex]\( x = 4 \)[/tex], we exclude this value from the domain. Thus, the domain includes all real numbers except [tex]\( x = 4 \)[/tex].

In interval notation, the domain is:
[tex]\[ (-\infty, 4) \cup (4, \infty) \][/tex]

Thus, the domain of [tex]\( f(x) \)[/tex] is [tex]\(\boxed{(-\infty, 4) \cup (4, \infty)}\)[/tex].