Let's solve the system of equations given by the functions [tex]\( f(x) = 4(2)^{-x} + 3 \)[/tex] and [tex]\( g(x) = 8 - 2^x \)[/tex].
We are required to find the solutions [tex]\(x\)[/tex] where [tex]\(f(x) = g(x)\)[/tex]. These solutions represent the values of [tex]\(x\)[/tex] where the two functions intersect.
Given:
1. [tex]\( f(x) = 4(2)^{-x} + 3 \)[/tex]
2. [tex]\( g(x) = 8 - 2^x \)[/tex]
We need to solve the equation:
[tex]\[ 4(2)^{-x} + 3 = 8 - 2^x \][/tex]
The solutions to this equation, [tex]\( x \)[/tex], are the points where the graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect. From this information, we find the solutions:
[tex]\[ x_1 = 0 \][/tex]
[tex]\[ x_2 = 2 \][/tex]
Next, we substitute these values into either [tex]\( f(x) \)[/tex] or [tex]\( g(x) \)[/tex] to find the corresponding [tex]\( y \)[/tex]-values:
For [tex]\( x_1 = 0 \)[/tex]:
[tex]\[ f(0) = 4(2)^{-0} + 3 = 4(1) + 3 = 4 + 3 = 7 \][/tex]
So, one solution is [tex]\( (0, 7) \)[/tex].
For [tex]\( x_2 = 2 \)[/tex]:
[tex]\[ f(2) = 4(2)^{-2} + 3 = 4\left(\frac{1}{4}\right) + 3 = 1 + 3 = 4 \][/tex]
So, another solution is [tex]\( (2, 4) \)[/tex].
In conclusion:
The solutions to the system of equations [tex]\( y = f(x) \)[/tex] and [tex]\( y = g(x) \)[/tex] are [tex]\( (0, 7) \)[/tex] and [tex]\( (2, 4) \)[/tex].
The solutions to the equation [tex]\( f(x) = g(x) \)[/tex] are [tex]\( 0 \)[/tex] and [tex]\( 2 \)[/tex].
