Question

NO LINKS!!!

55, Write an equation satisfying the given conditions.

Answer 1

Part (a)

The two limit statements tell us that this an exponential decay function.

The curve goes up forever when heading to the left (negative infinity) as indicated by the notation
`\displaystyle \lim_{\text{x}\to-\infty}f(x) = \infty`

At the same time, the curve slowly approaches the horizontal asymptote y = -2, when moving to the right, because of this notation
`\displaystyle \lim_{\text{x}\to\infty}f(x) = -2`

An exponential decay function like
`\text{y} = (0.5)^{\text{x}}` has a horizontal asymptote of y = 0, aka the x axis. The y value approaches 0 but never gets there. Each output is positive.

Shift everything down 2 units to arrive at
`\text{y} = (0.5)^{\text{x}}-2` and this will move the horizontal asymptote down the same amount.

There's nothing really special about the 0.5; you can replace it with any value in the interval 0 < b < 1.

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Answer:
`\text{f(x)} = (0.5)^{\text{x}}-2`

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Part (b)

I'll use this template

`\text{y} = ab^{\text{x}}+c`

Plugging in x = 0 leads to y = a+c which is the y intercept. Set this equal to the stated y intercept 7 and we get a+c = 7.

We want the
`ab^{\text{x}}` portion to approach zero, which leads to c = 4 so we approach the stated horizontal asymptote.

So,

a+c = 7

a+4 = 7

a = 7-4

a = 3

We go from this

`\text{y} = ab^{\text{x}}+c`

to this

`\text{y} = 3b^{\text{x}}+4`

The value of b doesn't matter.

I'll go for b = 0.7 so we get to
`\text{f(x)} = 3(0.7)^{\text{x}}+4`

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Answer:
`\text{g(x)} = 3(0.7)^{\text{x}}+4`

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Part (c)

The parent function
`\text{y} = \log(\text{x}})` has a domain of
`(0, \infty)`. In other words it is the interval
`0 < \text{x} < \infty`

If we replaced each input x with x-5, then we shift the xy axis 5 units to the left. It gives the illusion the log curve moves 5 units to the right.

The vertical asymptote also moves 5 units to the right. We go from a domain of
`(0, \infty)` to a domain of
`(5, \infty)`

The base of the log doesn't matter.

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Answer:
`\text{h(x)} = \log(\text{x}-5)`

Check out the graphs below. I used GeoGebra, but Desmos is another good option.