Answer:
8. Domain: (-∞, -15) ∪ (-15, -5) ∪ (-5, ∞)
9. Domain: [7/13, ∞)
Range: [1, ∞)
Step-by-step explanation:
<u>Question 8</u>
Given rational function:
![f(x)=\dfrac{x}{x^2+20x+75}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdfrac%7Bx%7D%7Bx%5E2%2B20x%2B75%7D)
Factor the denominator of the given rational function:
![\implies x^2+20x+75](https://tex.z-dn.net/?f=%5Cimplies%20x%5E2%2B20x%2B75)
![\implies x^2+5x+15x+75](https://tex.z-dn.net/?f=%5Cimplies%20x%5E2%2B5x%2B15x%2B75)
![\implies x(x+5)+15(x+5)](https://tex.z-dn.net/?f=%5Cimplies%20x%28x%2B5%29%2B15%28x%2B5%29)
![\implies (x+15)(x+5)](https://tex.z-dn.net/?f=%5Cimplies%20%28x%2B15%29%28x%2B5%29)
Therefore:
![f(x)=\dfrac{x}{(x+15)(x+5)}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdfrac%7Bx%7D%7B%28x%2B15%29%28x%2B5%29%7D)
<u>Asymptote</u>: a line that the curve gets infinitely close to, but never touches.
The function is <u>undefined</u> when the <u>denominator equals zero</u>:
![x+15=0 \implies x=-15](https://tex.z-dn.net/?f=x%2B15%3D0%20%5Cimplies%20x%3D-15)
![x+5=0 \implies x=-5](https://tex.z-dn.net/?f=x%2B5%3D0%20%5Cimplies%20x%3D-5)
Therefore, there are <u>vertical asymptotes</u> at x = -15 and x = -5.
<u>Domain</u>: set of all possible input values (x-values)
Therefore, the <u>domain of the given rational function</u> is:
(-∞, -15) ∪ (-15, -5) ∪ (-5, ∞)
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<u>Question 9</u>
Given function:
![f(x)=\sqrt{13x-7}+1](https://tex.z-dn.net/?f=f%28x%29%3D%5Csqrt%7B13x-7%7D%2B1)
<u>Domain</u>: set of all possible input values (x-values)
As the <u>square root of a negative number</u> is <u>undefined</u>:
![\implies 13x-7\geq 0](https://tex.z-dn.net/?f=%5Cimplies%2013x-7%5Cgeq%200)
![\implies 13x\geq 7](https://tex.z-dn.net/?f=%5Cimplies%2013x%5Cgeq%207)
![\implies x\geq \dfrac{7}{13}](https://tex.z-dn.net/?f=%5Cimplies%20x%5Cgeq%20%5Cdfrac%7B7%7D%7B13%7D)
Therefore, the <u>domain of the given function</u> is:
![\left[\dfrac{7}{13},\infty\right)](https://tex.z-dn.net/?f=%5Cleft%5B%5Cdfrac%7B7%7D%7B13%7D%2C%5Cinfty%5Cright%29)
<u>Range</u>: set of all possible output values (y-values)
![\textsf{As }\:\sqrt{13x-7}\geq 0](https://tex.z-dn.net/?f=%5Ctextsf%7BAs%20%7D%5C%3A%5Csqrt%7B13x-7%7D%5Cgeq%200)
![\implies \sqrt{13x-7}+1\geq 1](https://tex.z-dn.net/?f=%5Cimplies%20%5Csqrt%7B13x-7%7D%2B1%5Cgeq%201)
Therefore, the <u>range of the given function</u> is:
[1, ∞)