Determine the domain of the function f(x)f\left(x\right).f(x)=7x−3f\left(x\right)=\frac{7}{x-3}

( − ∞ , 3 ) ∪ ( 3 , ∞ ) \left(-\infty,3\right)\cup\left(3,\infty\right)

Determine the domain of the function f ( x ) f\left(x\right) . f ( x ) = 7 x − 3 f\left(x\right)=\frac{7}{x-3}

this problem, we are asked to determine the domain of our function f of x. And notice we have the rational function seven over x minus three. That's what our f of x is. Now I know when it comes to rational functions, the denominator that we have cannot be equal to zero. So let's go ahead and set this up to see if we can figure out what restrictions we would have on x. So I'm gonna solve this equation right here by first adding three on both sides of the equation. That'll get the threes to cancel, leaving me with x cannot be equal to three. So plugging in three is not allowed because that would make the denominator zero. So if I want to write this domain in interval notation, well, that's going to be from negative infinity all the way up to three not included, so we use a parenthesis, and then from three not included all the way up to infinity. So basically, all real numbers except for three will be allowed, and that would be the solution for this domain. 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 ) = 7 x − 3 f\left(x\right)=\frac{7}{x-3}

( − ∞ , 3 ) ∪ ( 3 , ∞ ) \left(-\infty,3\right)\cup\left(3,\infty\right)

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

( 3 , ∞ ) \left(3,\infty\right)

( − ∞ , 3 ] ∪ [ 3 , ∞ ) \left(-\infty,3\right\rbrack\cup\left\lbrack3,\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 $f (x)$ .
$f (x) = \frac{7}{x - 3}$

$(- \infty, \infty)$

$open paren 3 comma infinity close paren$

$open paren negative infinity comma 3 close paren union open paren 3 comma infinity close paren$

$open paren negative infinity comma 3 close bracket union open bracket 3 comma infinity close paren$

Verified step by step guidance

Identify the function given: $f\left(x\right) = \frac{7}{x - 3}$. This is a rational function where the denominator cannot be zero because division by zero is undefined.

Set the denominator equal to zero to find values that are not allowed in the domain: $x - 3 = 0$.

Solve the equation for $x$: $x = 3$. This means $x = 3$ is excluded from the domain.

Express the domain as all real numbers except $x = 3$. In interval notation, this is written as $\left(-\infty, 3\right) \cup \left(3, \infty\right)$.

Conclude that the domain includes all real numbers except where the denominator is zero, so the function is defined for every $x$ except $x = 3$.

5:34m

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