Mathematics, 14.07.2019 18:00 ashleyprescot05

Write an equation that has a hole at x = 2, a hole at x = 0 and a vertical asymptote at x = 1.reply i really don't understand how to do this and my grade depends on this

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Anytime you have a hole in a rational function, it means that both the numerator and denominator must have zero as that value. So if the hole is at x = 2, it means that when you evaluate the rational function at 2, there's going to be a zero in the numerator and denominator.

What makes it so that when evaluate at 2 to get a zero? x - 2. This comes from x = 2 and subtracting 2 from both sides. Put 2 into x-2 and we get the zero we want. So $\frac{x-2}{x-2}$ must be in the function.

The next discontinuity is a hole at zero. That's the same process as above, and x will work. Put in 0, get 0 back. So $\frac{x}{x}$ must be there too. We are going to MULTIPLY both pieces we have so we make sure both are there.

Lastly is the vertical asymptote at x = 1. The denominator must be zero, the numerator here can be anything. We set x = 1 equal to zero and subtract one from both sides to give x-1. Our function needs $\frac{1}{x-1}$ to include the asymptote. We use a 1 here because it's just a placeholder. Any number would work or any variable would work as long as it wasn't x-1.

Now we put them together and MULTIPLY all the pieces.

$y = \frac{x-2}{x-2} * \frac{x}{x} * \frac{1}{x-1}$ has all the pieces - the hole at 2, the hole at 0, and the vertical asymptote at 1. (We can multiply all the pieces out if we want).

Answer from: Quest

you didn't post any answer choices, so its not 100% clear what your teacher is looking for

however, log(15/7) is equivalent to log(15) - log(7)

i'm using the rule log(x/y) = log(x) - log(y)

Answer from: Quest

the solution of the equation is 2.23 ⇒ answer b

step-by-step explanation:

* lets revise the meaning of exponential function

- the form of the exponential function is y = ab^x, where a ≠ 0, b > 0 ,

b ≠ 1, and x is any real number

- it has a constant base b

- it has a variable exponent x

- to solve an exponential equation, take the log or ln of both sides,

and solve for the variable

* lets solve the problem

∵ $7^{x}=77$

- the base is 7 and the exponent is x

- insert ㏑ in both sides

∴ $ln(7^{x})=ln(77)$

- use the rule $ln(a^{n})= nln(a)$

∴ x ㏑(7) = ㏑(77)

- to find x divide both sides by ㏑(7)

∴ $x=\frac{ln(77)}{ln(7)}=2.23$

* the solution of the equation is 2.23 to the nearest 2 decimal places

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Write an equation that has a hole at x = 2, a hole at x = 0 and a vertical asymptote at x = 1.reply...

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