Answer:
Initial value (a) = 1
Base (b) = ¹/₃
Domain = (-∞, ∞)
Step-by-step explanation:
General form of an exponential function:
[tex]y=ab^x[/tex]
where:
- a is the initial value (y-intercept)
- b is the base (growth/decay factor) in decimal form
- x is the independent variable
- y is the dependent variable
If b > 1 then it is an increasing function.
If 0 < b < 1 then it is a decreasing function.
Initial value (a)
The initial value is the y-intercept.
The y-intercept is where the curve crosses the y-axis (when x = 0).
We have been given the point (0, 1), therefore the y-intercept is 1, and so:
Base (b)
To find the value of the base (b), substitute the found value of a and one of the given points (2, ¹/₉) into the equation:
[tex]\implies y=ab^x[/tex]
[tex]\implies \dfrac{1}{9}=(1)b^2[/tex]
[tex]\implies b^2=\dfrac{1}{9}[/tex]
[tex]\implies \sqrt{b^2}=\sqrt{\dfrac{1}{9}}[/tex]
[tex]\implies b=\dfrac{1}{3}[/tex]
Therefore:
Finally, the equation of the given exponential function is:
[tex]y=\left(\dfrac{1}{3}\right)^x[/tex]
Domain
The domain of a function is the set of all possible input values (x-values).
The domain of the found exponential function is unrestricted.
Therefore:
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