The graphs of [tex]$f(x) = 4(2)^{-x} + 3$[/tex] and [tex]$g(x) = 8 - 2^x$[/tex] are shown.

Place numbers in the blanks to complete the sentences.

The solutions to the system of equations [tex][tex]$y = f(x)$[/tex][/tex] and [tex]$y = g(x)$[/tex] are ( [tex]$\qquad$[/tex] , [tex]$\qquad$[/tex] ) and ( [tex]$\qquad$[/tex] , [tex]$\qquad$[/tex] ). The solutions to the equation [tex]$f(x) = g(x)$[/tex] are [tex]$\qquad$[/tex] and [tex]$\qquad$[/tex] .



Answer :

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].