To determine the range of the function [tex]\( f(x) = \frac{1}{7}(9)^x \)[/tex], follow these steps:
1. Understand the Form of the Function: The given function is an exponential function with the form [tex]\( f(x) = \frac{1}{7} \cdot 9^x \)[/tex]. Here, [tex]\( 9^x \)[/tex] is an exponential expression where the base is 9 and the exponent is [tex]\( x \)[/tex]. The coefficient [tex]\( \frac{1}{7} \)[/tex] scales the output of [tex]\( 9^x \)[/tex].
2. Exponential Functions Overview: For an exponential function [tex]\( a \cdot b^x \)[/tex] where [tex]\( b > 1 \)[/tex] and [tex]\( a > 0 \)[/tex], the range is all positive real numbers. This is because [tex]\( b^x \)[/tex] can take any positive value as [tex]\( x \)[/tex] varies over all real numbers. Thus, [tex]\( a \cdot b^x \)[/tex] will also take any positive value since [tex]\( a \)[/tex] is a positive constant.
3. Analyze the Base and Coefficient: In our function, the base [tex]\( b = 9 \)[/tex] is greater than 1, and the coefficient [tex]\( a = \frac{1}{7} \)[/tex] is positive. This means that as [tex]\( x \)[/tex] varies over all real numbers, [tex]\( 9^x \)[/tex] will produce all positive real numbers. Multiplying these values by [tex]\( \frac{1}{7} \)[/tex] will still yield all positive real numbers because a positive constant times a positive number is positive.
4. Range Determination: Since [tex]\( \frac{1}{7} \cdot 9^x \)[/tex] results in all positive values for any real number [tex]\( x \)[/tex], the range of the function [tex]\( f(x) \)[/tex] is all positive real numbers.
Therefore, the range of the function [tex]\( f(x) = \frac{1}{7}(9)^x \)[/tex] is:
all real numbers greater than 0.
Hence, the correct option is:
- all real numbers greater than 0
