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Likurg_2 [28]
1 year ago
11

8. What is the domain of

="\frac{x}{x^{2} +20x+75}" alt="\frac{x}{x^{2} +20x+75}" align="absmiddle" class="latex-formula"> ? Hint: try factoring the polynomial.
9. What is the domain and range of \sqrt{13x-7}+1 ?

Please hurry! I really need help with this!
Mathematics
1 answer:
antiseptic1488 [7]1 year ago
8 0

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}

Factor the denominator of the given rational function:

\implies x^2+20x+75

\implies x^2+5x+15x+75

\implies x(x+5)+15(x+5)

\implies (x+15)(x+5)

Therefore:

f(x)=\dfrac{x}{(x+15)(x+5)}

<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

x+5=0 \implies x=-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, ∞)

---------------------------------------------------------------------------------

<u>Question 9</u>

Given function:

f(x)=\sqrt{13x-7}+1

<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

\implies 13x\geq 7

\implies x\geq \dfrac{7}{13}

Therefore, the <u>domain of the given function</u> is:

\left[\dfrac{7}{13},\infty\right)

<u>Range</u>: set of all possible output values (y-values)

\textsf{As }\:\sqrt{13x-7}\geq 0

\implies \sqrt{13x-7}+1\geq 1

Therefore, the <u>range of the given function</u> is:

[1, ∞)

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