To find the domain and range of the function [tex]\((g \circ f)(x)\)[/tex], we need to first understand the individual functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Given:
[tex]\[ f(x) = e^x \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
1. Domain of [tex]\(f(x)\)[/tex]:
- The function [tex]\(f(x) = e^x\)[/tex] is an exponential function.
- Exponential functions have a domain of all real numbers, because you can take the exponent of any real number.
- Therefore, the domain of [tex]\(f(x)\)[/tex] is all real numbers.
2. Applying [tex]\(f(x)\)[/tex] to [tex]\(g(x)\)[/tex]:
- We need to find [tex]\( (g \circ f)(x) \)[/tex], which is [tex]\( g(f(x)) \)[/tex].
- Substituting [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex], we get:
[tex]\[ g(f(x)) = g(e^x) = e^x + 6 \][/tex]
3. Domain of [tex]\((g \circ f)(x)\)[/tex]:
- Since [tex]\( f(x) = e^x \)[/tex] can take any real number as its input, the output of [tex]\( f(x) \)[/tex] is [tex]\( e^x \)[/tex].
- Similarly, [tex]\( g(x) = x + 6 \)[/tex] can take any real number as its input.
- Therefore, the function [tex]\( (g \circ f)(x) = e^x + 6 \)[/tex] can take any real input [tex]\(x\)[/tex] without restriction.
- Thus, the domain of [tex]\( (g \circ f)(x) \)[/tex] is all real numbers.
4. Range of [tex]\(f(x)\)[/tex]:
- The function [tex]\( f(x) = e^x \)[/tex] has an output that is always positive, ranging from [tex]\(0\)[/tex] (exclusive) to [tex]\(\infty\)[/tex].
- Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex].
5. Range of [tex]\((g \circ f)(x)\)[/tex]:
- Now, [tex]\( g(x) = x + 6 \)[/tex] shifts the output of [tex]\( f(x) \)[/tex] by +6 units.
- Since [tex]\( f(x) \)[/tex] outputs values in the range [tex]\((0, \infty)\)[/tex]:
[tex]\[ g(f(x)) = e^x + 6 \][/tex]
- Shifting the range [tex]\((0, \infty)\)[/tex] by 6 units results in the new range [tex]\((6, \infty)\)[/tex], because adding 6 to any positive number greater than 0 will yield a result greater than 6.
- Therefore, the range of [tex]\( (g \circ f)(x) \)[/tex] is [tex]\( y > 6 \)[/tex].
In summary, the domain of [tex]\((g \circ f)(x)\)[/tex] is all real numbers, and the range is [tex]\( y > 6 \)[/tex].
Thus, the correct answer is:
[tex]\[ \text{domain: all real numbers, range: } y > 6 \][/tex]
