The function [tex]\( f(x) = 9^x - 8 \)[/tex] can be derived from the function [tex]\( g(x) = 9^x \)[/tex] by applying a transformation. Here's how we can analyze the situation step-by-step:
### Transformation for the Graph
In the function [tex]\( f(x) = 9^x - 8 \)[/tex], the term [tex]\(-8\)[/tex] indicates a vertical shift downward by 8 units. The value of the function [tex]\( f(x) \)[/tex] is the same as [tex]\( g(x) \)[/tex] but decreased by a constant value of 8 for any given [tex]\( x \)[/tex]. Hence:
- The correct answer for the transformation is:
[tex]\[ (d) \text{ shifting the graph of } g(x) \text{ downward 8 units} \][/tex]
Your answer: [tex]\( \boxed{d} \)[/tex]
### Domain of the Function
The domain of [tex]\( g(x) = 9^x \)[/tex] is all real numbers, [tex]\( (-\infty, \infty) \)[/tex]. Subtracting 8 from [tex]\( g(x) \)[/tex] does not affect the domain, hence the domain of [tex]\( f(x) = 9^x - 8 \)[/tex]:
- The domain remains [tex]\( (-\infty, \infty) \)[/tex].
Your answer: [tex]\( \boxed{Yes} \)[/tex]
### Range of the Function
For the function [tex]\( g(x) = 9^x \)[/tex], the range is [tex]\( (0, \infty) \)[/tex] since [tex]\( 9^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex].
When we transform [tex]\( g(x) \)[/tex] by subtracting 8 to form [tex]\( f(x) \)[/tex], the range shifts accordingly. Since the smallest value [tex]\( g(x) \)[/tex] can approach is 0 (but not including 0), the smallest value [tex]\( f(x) \)[/tex] will approach is [tex]\( 0 - 8 = -8 \)[/tex]. Therefore, the range of [tex]\( f(x) \)[/tex] becomes [tex]\( (-8, \infty) \)[/tex].
- The minimum value, [tex]\( A \)[/tex], in the range [tex]\( (A, \infty) \)[/tex] is:
[tex]\[ A = -8 \][/tex]
Your answer: [tex]\( \boxed{-8} \)[/tex]
