Find the domain of the rational function. Then, write it in lowest terms.f(x)=6x52x2−8f\left(x\right)=\frac{6x^5}{2x^2-8}f(x)=2x2−86x5

{ x ∣ x ≠ 2 , − 2 } , f ( x ) = 3 x 5 x 2 − 4 {\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{3x^5}{x^2-4} { x ∣ x  = 2 , − 2 } , f ( x ) = x 2 − 4 3 x 5

Find the domain of the rational function. Then, write it in lowest terms. f ( x ) = 6 x 5 2 x 2 − 8 f\left(x\right)=\frac{6x^5}{2x^2-8} f ( x ) = 2 x 2 − 8 6 x 5

In this problem, we're asked to find the domain of our rational function and then write it in lowest terms. Remember, we always want to find our domain before writing in lowest terms, so let's go ahead and start there. Now we want to take our denominator here, which is 2x squared minus 8, and set it equal to 0 to find our domain. So solving for x here, I'm going to go ahead and add 8 to both sides, leaving me with 2x squared is equal to 8. Then I'm going to go ahead and divide both sides by 2, leaving me with x squared equals 4. And then finally, my last step here in isolating x is taking the square root of both sides, which will leave me with x is equal to plus or minus the square root of 4, which we know is just 2. So this tells me that I have two numbers in my domain restriction, two numbers that would make my denominator 0 that x cannot be. So my domain is x such that x cannot be equal to positive 2 or negative 2, and there's my domain. Now that we have our domain, let's go ahead and focus on writing our function in lowest terms. So I'm going to go ahead and copy my function again down here just so that we can start fresh. F of x is equal to 6x to the 5th divided by 2x squared minus 8. Okay. Remember, we want to factor both the top and the bottom of our function, so let's go ahead and do that. Now for the numerator, it's not immediately obvious to me what I can factor here, so I'm gonna go ahead and start with the denominator. Now in the denominator, I see that I have a greatest common factor between these two terms of 2. So I'm gonna go ahead and pull out a 2 from each of them, leaving me with 2 times x squared minus 4. Now looking at my numerator, I see that I can also go ahead and pull out a 2 there because I know that that will cancel. So I'm gonna go ahead and pull out a 2 of my numerator as well. So pulling out that 2 leaves me with 2 times 3x to the power of 5. Now for my numerator, it doesn't look like I can factor anything else, and I could factor my denominator more, this x squared minus 4. I could actually factor into x+2 timesxminus2, but I can tell that that's not going to cancel with anything in my numerator so it's not really necessary here. So I'm just going to go ahead and cancel this 2 out of my function and that will leave me with 3x to the 5th as my numerator divided by x squared minus 4 and that is my function in lowest terms. 3x to the 5th divided by x squared minus 4 and we're done here. I'll see you in the next video.

Find the domain of the rational function. Then, write it in lowest terms. f ( x ) = 6 x 5 2 x 2 − 8 f\left(x\right)=\frac{6x^5}{2x^2-8} f ( x ) = 2 x 2 − 8 6 x 5

{ x ∣ x ≠ 2 , − 2 } , f ( x ) = 3 x 5 x 2 − 4 {\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{3x^5}{x^2-4} { x ∣ x  = 2 , − 2 } , f ( x ) = x 2 − 4 3 x 5

{ x ∣ x ≠ 2 } , f ( x ) = 3 x 5 x 2 − 4 {\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-4} { x ∣ x  = 2 } , f ( x ) = x 2 − 4 3 x 5

{ x ∣ x ≠ 2 } , f ( x ) = 3 x 5 x 2 − 8 {\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-8} { x ∣ x  = 2 } , f ( x ) = x 2 − 8 3 x 5

0. Functions

Common Functions

Business Calculus

0. Functions

Common Functions

Struggling with Business Calculus?

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Multiple Choice

Find the domain of the rational function. Then, write it in lowest terms.
$f\left(x\right)=\frac{6x^5}{2x^2-8}$

${\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{3x^5}{x^2-4}$

${\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{6x^5}{2x^2-8}$

${\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-4}$

${\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-8}$

Verified step by step guidance

Step 1: Identify the domain of the rational function. The domain of a rational function is all real numbers except where the denominator equals zero. For the given function f(x) = 6x^5 / (2x^2 - 8), set the denominator 2x^2 - 8 equal to zero and solve for x.

Step 2: Solve the equation 2x^2 - 8 = 0. First, factor out the common term 2 from the denominator: 2(x^2 - 4) = 0. Then, solve x^2 - 4 = 0 using the difference of squares formula: x^2 - 4 = (x - 2)(x + 2).

Step 3: Determine the values of x that make the denominator zero. From the factored form (x - 2)(x + 2) = 0, solve for x: x = 2 and x = -2. These are the values excluded from the domain.

Step 4: Simplify the rational function to its lowest terms. Factor the numerator and denominator if possible. The numerator is 6x^5, and the denominator is 2(x^2 - 4). Simplify by dividing both the numerator and denominator by their greatest common factor, which is 2.

Step 5: Write the simplified function and the domain. The simplified function is f(x) = 3x^5 / (x^2 - 4), and the domain is all real numbers except x = 2 and x = -2. In set notation, the domain is {x | x ≠ 2, -2}.

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