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?
What are the solutions of the equation −6x+14=−2^x+6?
Answers
Answer
Theres multiple correct answers
Answered by
GPT-4o mini
To find the solutions to the equation \( -6x + 14 = -2^x + 6 \), we can start by rewriting the equation to isolate terms:
\[
-6x + 14 + 2^x - 6 = 0
\]
This simplifies to:
\[
-6x + 2^x + 8 = 0
\]
This equation can be graphical solved by plotting the two functions \( f(x) = -6x + 14 \) and \( g(x) = -2^x + 6 \) and finding their intersection points.
### Steps to Graph the Functions:
1. **Graph \( f(x) = -6x + 14 \)**:
- This is a linear function with a slope of \(-6\) and a y-intercept of \(14\).
- When \( x = 0 \), \( f(0) = 14 \).
- When \( x = 2 \), \( f(2) = -12 + 14 = 2 \).
- When \( x = \frac{14}{6} \approx 2.33 \), \( f(x) \) crosses the x-axis.
2. **Graph \( g(x) = -2^x + 6 \)**:
- This is an exponential function.
- When \( x = 0 \), \( g(0) = -1 + 6 = 5 \).
- As \( x \) increases, \( 2^x \) becomes larger leading \( g(x) \) to decrease (since it has a negative sign).
- At \( x = 2 \), \( g(2) = -4 + 6 = 2 \).
- As \( x \) approaches positive infinity, \( g(x) \) approaches \( -\infty \).
### Finding the Intersection Point:
To find the solutions algebraically, you can proceed to find approximate roots through numerical methods or graphically as described.
From our graphical assessment:
- At \( x = 2 \), both functions yield the same value of \( 2 \): \( f(2) = 2 \) and \( g(2) = 2 \).
To confirm if there are any other intersections, we can check values around \( x = 2 \) and perhaps solve numerically using methods like the graphical method or Newton's method to approximate other roots if they exist.
### Conclusion:
1. The one obvious intersection is at \( x = 2 \).
2. Based on the nature of these functions (linear and exponential), there is likely only this single solution.
Therefore, the solution to the equation \( -6x + 14 = -2^x + 6 \) is:
\[
\boxed{2}
\]
If further verification is needed or capturing other potential solutions, numeric methods or software tools can be used to draw a complete graph of the functions \( f(x) \) and \( g(x) \) for more visual confirmation.
\[
-6x + 14 + 2^x - 6 = 0
\]
This simplifies to:
\[
-6x + 2^x + 8 = 0
\]
This equation can be graphical solved by plotting the two functions \( f(x) = -6x + 14 \) and \( g(x) = -2^x + 6 \) and finding their intersection points.
### Steps to Graph the Functions:
1. **Graph \( f(x) = -6x + 14 \)**:
- This is a linear function with a slope of \(-6\) and a y-intercept of \(14\).
- When \( x = 0 \), \( f(0) = 14 \).
- When \( x = 2 \), \( f(2) = -12 + 14 = 2 \).
- When \( x = \frac{14}{6} \approx 2.33 \), \( f(x) \) crosses the x-axis.
2. **Graph \( g(x) = -2^x + 6 \)**:
- This is an exponential function.
- When \( x = 0 \), \( g(0) = -1 + 6 = 5 \).
- As \( x \) increases, \( 2^x \) becomes larger leading \( g(x) \) to decrease (since it has a negative sign).
- At \( x = 2 \), \( g(2) = -4 + 6 = 2 \).
- As \( x \) approaches positive infinity, \( g(x) \) approaches \( -\infty \).
### Finding the Intersection Point:
To find the solutions algebraically, you can proceed to find approximate roots through numerical methods or graphically as described.
From our graphical assessment:
- At \( x = 2 \), both functions yield the same value of \( 2 \): \( f(2) = 2 \) and \( g(2) = 2 \).
To confirm if there are any other intersections, we can check values around \( x = 2 \) and perhaps solve numerically using methods like the graphical method or Newton's method to approximate other roots if they exist.
### Conclusion:
1. The one obvious intersection is at \( x = 2 \).
2. Based on the nature of these functions (linear and exponential), there is likely only this single solution.
Therefore, the solution to the equation \( -6x + 14 = -2^x + 6 \) is:
\[
\boxed{2}
\]
If further verification is needed or capturing other potential solutions, numeric methods or software tools can be used to draw a complete graph of the functions \( f(x) \) and \( g(x) \) for more visual confirmation.
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