Question

The graphs of f of x equals 4 left-parenthesis 2 right-parenthesis Superscript negative x Baseline plus 3 and g of x equals 8 minus 2 Superscript x Baseline are shown.

A graph.Short description, A graph.,Long description,
On a coordinate plane, the graphs of 2 functions are shown: f of x and g of x.

F of x is curve that opens down and left. It crosses the y-axis at (0, 7), passes through point (2, 4), and crosses the x-axis at (3, 0).
G of x is a curve that opens up and right. It crosses the y-axis at (0, 7), passes through point (2, 4), and does not cross the x-axis.

Question
Place numbers in the blanks to complete the sentences.

Response area with 6 blank spaces
The solutions to the system of equations y equals f of x and y equals g of x are left-parenthesis
Blank space 1 empty
comma
Blank space 2 empty
right-parenthesis and left-parenthesis
Blank space 7 empty
comma
Blank space 4 empty
right-parenthesis. The solutions to the equation f of x equals g of x are
Blank space 5 empty
and
Blank space 6 empty
.
Answer options with 6 options.

Answer 1

To find the solutions of the system of equations \( y = f(x) \) and \( y = g(x) \), we need to set \( f(x) = g(x) \):

1. The functions are:
- \( f(x) = 4(2^{-x}) + 3 \)
- \( g(x) = 8 - 2^x \)

2. Let's solve the equation \( 4(2^{-x}) + 3 = 8 - 2^x \):

First, we rewrite \( 2^{-x} \) as \( \frac{1}{2^x} \):
\[
4\left(\frac{1}{2^x}\right) + 3 = 8 - 2^x
\]
\[
\frac{4}{2^x} + 3 = 8 - 2^x
\]

Next, create a common equation by multiplying both sides by \( 2^x \) (to eliminate the fraction):
\[
4 + 3 \cdot 2^x = 8 \cdot 2^x - (2^x)^2
\]

Rearranging this gives:
\[
(2^x)^2 - 5\cdot2^x + 4 = 0
\]

Let \( u = 2^x \):
\[
u^2 - 5u + 4 = 0
\]

Solving this quadratic equation:
\[
u = \frac{5 \pm \sqrt{(5)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}
\]
\[
u = \frac{5 \pm \sqrt{25 - 16}}{2}
\]
\[
u = \frac{5 \pm 3}{2}
\]

Thus, we have two solutions for \( u \):
- \( u = \frac{8}{2} = 4 \)
- \( u = \frac{2}{2} = 1 \)

3. Now we can convert back to \( x \) for both solutions:
- If \( u = 4 \), then \( 2^x = 4 \) implies \( x = 2 \).
- If \( u = 1 \), then \( 2^x = 1 \) implies \( x = 0 \).

4. Corresponding y-values (using either function, e.g., \( f(x) \)):
- For \( x = 2 \): \( f(2) = 4(2^{-2}) + 3 = 4(0.25) + 3 = 1 + 3 = 4 \).
- For \( x = 0 \): \( f(0) = 4(2^0) + 3 = 4(1) + 3 = 4 + 3 = 7 \).

So the solutions are:
- \( (0, 7) \)
- \( (2, 4) \)

Now, we can fill in the blanks:

- Blank space 1: 0
- Blank space 2: 7
- Blank space 3: 2
- Blank space 4: 4
- Blank space 5: 0
- Blank space 6: 7

Thus, the complete sentences should read:
The solutions to the system of equations \( y = f(x) \) and \( y = g(x) \) are \( (0, 7) \) and \( (2, 4) \). The solutions to the equation \( f(x) = g(x) \) are \( 0 \) and \( 7 \).