To evaluate the given functions [tex]\( f(x) = 2^x \)[/tex] and [tex]\( g(x) = \left(\frac{1}{3}\right)^x \)[/tex], let's go through each step systematically.
1. Evaluating [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = \left(\frac{1}{3}\right)^0 = 1.0 \][/tex]
2. Evaluating [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 2^0 = 1 \][/tex]
3. Evaluating [tex]\( g(-3) \)[/tex]:
[tex]\[ g(-3) = \left(\frac{1}{3}\right)^{-3} = 27.000000000000004 \][/tex]
4. Evaluating [tex]\( g(-1) \)[/tex]:
[tex]\[ g(-1) = \left(\frac{1}{3}\right)^{-1} = 3.0 \][/tex]
5. Evaluating [tex]\( f(-1) \)[/tex]:
[tex]\[ f(-1) = 2^{-1} = 0.5 \][/tex]
6. Evaluating [tex]\( f(-4) \)[/tex]:
[tex]\[ f(-4) = 2^{-4} = 0.0625 \][/tex]
7. Evaluating [tex]\( f(2) + g(-2) \)[/tex]:
[tex]\[ f(2) + g(-2) = 2^2 + \left(\frac{1}{3}\right)^{-2} = 4 + 9 = 13.000000000000002 \][/tex]
8. Evaluating [tex]\( f(2) - g(-2) \)[/tex]:
[tex]\[ f(2) - g(-2) = 2^2 - \left(\frac{1}{3}\right)^{-2} = 4 - 9 = -5.000000000000002 \][/tex]
So, in summary:
1. [tex]\( g(0) = 1.0 \)[/tex]
2. [tex]\( f(0) = 1 \)[/tex]
3. [tex]\( g(-3) = 27.000000000000004 \)[/tex]
4. [tex]\( g(-1) = 3.0 \)[/tex]
5. [tex]\( f(-1) = 0.5 \)[/tex]
6. [tex]\( f(-4) = 0.0625 \)[/tex]
7. [tex]\( f(2) + g(-2) = 13.000000000000002 \)[/tex]
8. [tex]\( f(2) - g(-2) = -5.000000000000002 \)[/tex]
