Which of the following functions is equivalent to the function below?
Select three that apply.
f(x)=(13)2x
g(x)=3−2x
g(x)=(16)x
g(x)=(23)−x
g(x)=3x2
g(x)=9−x
To determine which functions are equivalent to f(x) = (13)2x, we need to simplify f(x) and compare it to the given options.
First, let's simplify f(x) = (13)2x. Using the exponent rule (a^b)^c = a^(b*c), we can simplify:
f(x) = (13)2x
= 1^2x * 3^(2x)
= 3^(2x)
Now, let's compare this simplified form of f(x) to the given options:
g(x) = 3−2x
This function is not equivalent to f(x) since it involves subtraction, whereas f(x) only has exponentiation.
g(x) = (16)x
This function is not equivalent to f(x) since it involves a constant base of 16, whereas f(x) has a base of 3.
g(x) = (23)−x
This function is equivalent to f(x) since it has the same base, 3, and the exponent -x is equivalent to 2x since (-x) = -1 * x.
g(x) = 3x2
This function is not equivalent to f(x) since it has the exponent 2x instead of 2x.
g(x) = 9−x
This function is not equivalent to f(x) since it involves subtraction, whereas f(x) only has exponentiation.
So, the functions that are equivalent to f(x) = (13)2x are:
- g(x) = (23)−x
Thus, the correct answer is one option: g(x) = (23)−x.
To determine which of the given functions is equivalent to f(x)=(13)2x, we can compare the expressions.
The given options are:
g(x)=3−2x
g(x)=(16)x
g(x)=(23)−x
g(x)=3x2
g(x)=9−x
To find the equivalent function, we need to ensure that the base and exponent are the same in both functions.
In f(x)=(13)2x, the base is 13 and the exponent is 2x.
Looking at the options, we see that g(x)=(16)x has a base of 16, which is not the same as 13. Therefore, it is not equivalent to f(x).
Similarly, g(x)=(23)−x has a base of 23, which is also not the same as 13.
Next, g(x)=3x2 has a base of 3, which is still not the same as 13.
Now let's consider g(x)=3−2x. In this function, the base is 3 and the exponent is -2x. Since the base is the same as f(x), we need to check if the exponents are equivalent.
To compare the exponents, we can rewrite f(x) as:
f(x)=2x(13)
Comparing the exponents, we can see that -2x is equal to 2x. The negative sign in front of the exponent means that the bases need to be reciprocals of each other. Therefore, g(x)=3−2x is equivalent to f(x)=(13)2x.
Finally, let's consider g(x)=9−x. In this function, the base is 9 and there is no exponent. This means that the exponent is 1. Since the exponent in f(x) is 2x, g(x)=9−x is not equivalent to f(x)=(13)2x.
Therefore, the three functions that are equivalent to f(x)=(13)2x are:
1. g(x)=3−2x
2. g(x)=3−2x
3. g(x)=(23)−x