Let's go through these answer choices one at a time to see which are true and which are false.
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Choice A) The y intercept is (0,3)
This is true because plugging in x = 0 leads to y = 3 as shown below
y = 3*(9)^x
y = 3*(9)^0
y = 3*1
y = 3
note how any nonzero number to the zeroth power is 1. Algebraically, the rule is x^0 = 1 where x is any number you want except 0.
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Choice B) The y intercept is (0,1)
This is false. We've shown in part A that the y intercept is (0,3). There can only be one y intercept for any function. Otherwise, it would fail the vertical line test. Furthermore, plugging x = 0 into the function does not lead to y = 1.
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Choice C) The domain of f(x) is x > 0
This is false because plugging in x = 0 leads to y = 3. In other words, x = 0 is in the domain of f(x). The same applies to any negative x value as well.
The actual domain in this case is the set of all real numbers. There are no restrictions on what to plug in (ie there are no division by zero errors to worry about for instance). Plugging in any real number for x leads to some real number for y.
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Choice D) It is always decreasing
This is false.
Plug in x = 0 and it leads to y = 3. Plug in x = 1 and it leads to...
y = 3*(9)^x
y = 3*(9)^1
y = 3*(9)
y = 27
So the two points (0,3) and (1,27) are on this f(x) function curve. The y values increase as we move from left to right (from x = 0 to x = 1). Therefore, the function is increasing on this interval. This contradicts the statement the function is always decreasing. In fact, it turns out that this particular function is always increasing for any interval you pick.
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<h3>Final Answer: Choice A) the y intercept is (0,3)</h3>
