Question
Graph the functions f(x)=−6x+14 and g(x)=−2^x+6 on the same coordinate plane.
What are the solutions of the equation −6x+14=−2^x+6? Theres multiple solutions-
Responses
x = 0
x = 0
x = 1
x = 1
x = 2
x = 2
x = 3
x = 3
x = 4
What are the solutions of the equation −6x+14=−2^x+6? Theres multiple solutions-
Responses
x = 0
x = 0
x = 1
x = 1
x = 2
x = 2
x = 3
x = 3
x = 4
Answers
GPT-4o mini
To solve the equation \( -6x + 14 = -2^x + 6 \), we first rearrange it into a more manageable form:
\[
-6x + 14 + 2^x - 6 = 0
\]
\[
-6x + 2^x + 8 = 0
\]
Now, we can analyze the functions \( f(x) = -6x + 14 \) (a linear function) and \( g(x) = -2^x + 6 \) (an exponential function) graphically.
1. **Graph \( f(x) = -6x + 14 \)**:
- This is a straight line with a slope of -6 and a y-intercept of 14. It decreases rapidly as \( x \) increases.
2. **Graph \( g(x) = -2^x + 6 \)**:
- This is an exponential decay function that starts at \( g(0) = 5 \) (because \( -2^0 + 6 = 5 \)) and approaches 6 as \( x \) approaches negative infinity, while it approaches \(-\infty\) as \( x \) increases.
Next, we can graph both functions on the same coordinate plane.
### Intersections of the Graphs
- The points where \( f(x) = g(x) \) represent the solutions to the equation \( -6x + 14 = -2^x + 6 \).
- By graphing, you would look for the x-values where the two functions intersect.
### Checking solution values:
To check the proposed solution values \( x = 0, 1, 2, 3, 4 \):
1. **For \( x = 0 \)**:
\[
f(0) = -6(0) + 14 = 14
\]
\[
g(0) = -2^0 + 6 = -1 + 6 = 5
\]
(Not an intersection)
2. **For \( x = 1 \)**:
\[
f(1) = -6(1) + 14 = 8
\]
\[
g(1) = -2^1 + 6 = -2 + 6 = 4
\]
(Not an intersection)
3. **For \( x = 2 \)**:
\[
f(2) = -6(2) + 14 = 2
\]
\[
g(2) = -2^2 + 6 = -4 + 6 = 2
\]
(Yes, an intersection!)
4. **For \( x = 3 \)**:
\[
f(3) = -6(3) + 14 = -4
\]
\[
g(3) = -2^3 + 6 = -8 + 6 = -2
\]
(Not an intersection)
5. **For \( x = 4 \)**:
\[
f(4) = -6(4) + 14 = -10
\]
\[
g(4) = -2^4 + 6 = -16 + 6 = -10
\]
(Yes, an intersection!)
### Summary of Solutions
From the calculations, the equation \( -6x + 14 = -2^x + 6 \) has solutions at:
- \( x = 2 \)
- \( x = 4 \)
**Thus, the solutions are \( x = 2 \) and \( x = 4 \).**
\[
-6x + 14 + 2^x - 6 = 0
\]
\[
-6x + 2^x + 8 = 0
\]
Now, we can analyze the functions \( f(x) = -6x + 14 \) (a linear function) and \( g(x) = -2^x + 6 \) (an exponential function) graphically.
1. **Graph \( f(x) = -6x + 14 \)**:
- This is a straight line with a slope of -6 and a y-intercept of 14. It decreases rapidly as \( x \) increases.
2. **Graph \( g(x) = -2^x + 6 \)**:
- This is an exponential decay function that starts at \( g(0) = 5 \) (because \( -2^0 + 6 = 5 \)) and approaches 6 as \( x \) approaches negative infinity, while it approaches \(-\infty\) as \( x \) increases.
Next, we can graph both functions on the same coordinate plane.
### Intersections of the Graphs
- The points where \( f(x) = g(x) \) represent the solutions to the equation \( -6x + 14 = -2^x + 6 \).
- By graphing, you would look for the x-values where the two functions intersect.
### Checking solution values:
To check the proposed solution values \( x = 0, 1, 2, 3, 4 \):
1. **For \( x = 0 \)**:
\[
f(0) = -6(0) + 14 = 14
\]
\[
g(0) = -2^0 + 6 = -1 + 6 = 5
\]
(Not an intersection)
2. **For \( x = 1 \)**:
\[
f(1) = -6(1) + 14 = 8
\]
\[
g(1) = -2^1 + 6 = -2 + 6 = 4
\]
(Not an intersection)
3. **For \( x = 2 \)**:
\[
f(2) = -6(2) + 14 = 2
\]
\[
g(2) = -2^2 + 6 = -4 + 6 = 2
\]
(Yes, an intersection!)
4. **For \( x = 3 \)**:
\[
f(3) = -6(3) + 14 = -4
\]
\[
g(3) = -2^3 + 6 = -8 + 6 = -2
\]
(Not an intersection)
5. **For \( x = 4 \)**:
\[
f(4) = -6(4) + 14 = -10
\]
\[
g(4) = -2^4 + 6 = -16 + 6 = -10
\]
(Yes, an intersection!)
### Summary of Solutions
From the calculations, the equation \( -6x + 14 = -2^x + 6 \) has solutions at:
- \( x = 2 \)
- \( x = 4 \)
**Thus, the solutions are \( x = 2 \) and \( x = 4 \).**