Determine the domain of the function f(x)f\left(x\right).f(x)=5xf\left(x\right)=\frac{5}{x}

( − ∞ , 0 ) ∪ ( 0 , ∞ ) \left(-\infty,0\right)\cup\left(0,\infty\right)

Determine the domain of the function f ( x ) f\left(x\right) . f ( x ) = 5 x f\left(x\right)=\frac{5}{x}

this problem, we are asked to determine the domain of the function f of x. And notice we have this rational function five over x. So how do we go about solving this problem? Well, the domain is basically telling us the restrictions that we have on the x values. It's what values for x we're allowed to have. Now one of the things I know when it comes to rational functions is we cannot have zero in the denominator. So what that means is x cannot be equal to zero. Now I want to represent this as my domain. And to write this as the domain, I'm going to use interval notation here. So I'm going to say we can go all the way from negative infinity up to zero, but we're not going to include that zero since x cannot equal zero. And then we'll have from zero all the way to infinity. So this is just telling us that we cannot include zero in our answer, but every other number is Okay. So all real numbers except for zero would be the solution to this problem. There are a few different ways you can represent the domain, but this is just how I'm going to do it. But any case where you have every number except zero as an answer, that's gonna be the situation and the solution for this problem. So hope you found this video helpful, and I'll see you all in the next one.

Determine the domain of the function f ( x ) f\left(x\right) . f ( x ) = 5 x f\left(x\right)=\frac{5}{x}

( − ∞ , 0 ) ∪ ( 0 , ∞ ) \left(-\infty,0\right)\cup\left(0,\infty\right)

( − ∞ , 0 ] ∪ [ 0 , ∞ ) \left(-\infty,0\right\rbrack\cup\left\lbrack0,\infty\right)

( − ∞ , ∞ ) \left(-\infty,\infty\right)

( 0 , ∞ ) \left(0,\infty\right)

8. Rational Expressions and Functions

Simplifying Rational Expressions

Intermediate Algebra

8. Rational Expressions and Functions

Simplifying Rational Expressions

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Multiple Choice

Determine the domain of the function $f (x)$ .
$f (x) = \frac{5}{x}$

$open paren negative infinity comma 0 close paren union open paren 0 comma infinity close paren$

$open paren negative infinity comma 0 close bracket union open bracket 0 comma infinity close paren$

$(- \infty, \infty)$

$open paren 0 comma infinity close paren$

Verified step by step guidance

Identify the function given: $f\left(x\right) = \frac{5}{x}$. This is a rational function where the denominator is $x$.

Recall that the domain of a function includes all real numbers except those that make the denominator zero, because division by zero is undefined.

Set the denominator equal to zero and solve for $x$: $x = 0$. This value is not allowed in the domain.

Express the domain as all real numbers except \$0$, which can be written in interval notation as $\left(-\infty, 0\right) \cup \left(0, \infty\right)$.

Conclude that the domain of $f\left(x\right)$ is all real numbers except \$0$, since the function is defined everywhere else.

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