### Finding the Intersection of Two Functions #### Problem Statement The functions given by \( f(x) = 7^{1-x} \) and \( g(x) = 5^{2-3x} \) are shown to intersect at a point as depicted in the figure below. #### Graphical Representation The graph presents the two functions \( f(x) = 7^{1-x} \) and \( g(x) = 5^{2-3x} \) plotted on a coordinate system. The x-axis ranges from 0 to 3, and the y-axis ranges from 0 to 10. 1. The function \( f(x) = 7^{1-x} \) is a decreasing exponential function. 2. The function \( g(x) = 5^{2-3x} \) is also a decreasing exponential function. The point of intersection is highlighted and labelled as (0.442, 2.964). #### Task Algebraically find the **EXACT** solution to \( f(x) = g(x) \) and then verify the three decimal approximation for the input value as the solution. #### Solution 1. Set the functions equal to each other: \[ 7^{1-x} = 5^{2-3x} \] 2. Solve for \( x \) by taking the natural logarithm of both sides. #### Graph Explanation - The black curve represents the function \( f(x) = 7^{1-x} \). - The blue curve represents the function \( g(x) = 5^{2-3x} \). - The intersection point at approximately \( x = 0.442 \) and \( y = 2.964 \) (marked in yellow) is where the two functions are equal. By solving the equation \( 7^{1-x} = 5^{2-3x} \) analytically, students can find the exact value of \( x \) that satisfies the equation.
### Finding the Intersection of Two Functions #### Problem Statement The functions given by \( f(x) = 7^{1-x} \) and \( g(x) = 5^{2-3x} \) are shown to intersect at a point as depicted in the figure below. #### Graphical Representation The graph presents the two functions \( f(x) = 7^{1-x} \) and \( g(x) = 5^{2-3x} \) plotted on a coordinate system. The x-axis ranges from 0 to 3, and the y-axis ranges from 0 to 10. 1. The function \( f(x) = 7^{1-x} \) is a decreasing exponential function. 2. The function \( g(x) = 5^{2-3x} \) is also a decreasing exponential function. The point of intersection is highlighted and labelled as (0.442, 2.964). #### Task Algebraically find the **EXACT** solution to \( f(x) = g(x) \) and then verify the three decimal approximation for the input value as the solution. #### Solution 1. Set the functions equal to each other: \[ 7^{1-x} = 5^{2-3x} \] 2. Solve for \( x \) by taking the natural logarithm of both sides. #### Graph Explanation - The black curve represents the function \( f(x) = 7^{1-x} \). - The blue curve represents the function \( g(x) = 5^{2-3x} \). - The intersection point at approximately \( x = 0.442 \) and \( y = 2.964 \) (marked in yellow) is where the two functions are equal. By solving the equation \( 7^{1-x} = 5^{2-3x} \) analytically, students can find the exact value of \( x \) that satisfies the equation.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Finding the Intersection of Two Functions
#### Problem Statement
The functions given by \( f(x) = 7^{1-x} \) and \( g(x) = 5^{2-3x} \) are shown to intersect at a point as depicted in the figure below.
#### Graphical Representation
The graph presents the two functions \( f(x) = 7^{1-x} \) and \( g(x) = 5^{2-3x} \) plotted on a coordinate system. The x-axis ranges from 0 to 3, and the y-axis ranges from 0 to 10.
1. The function \( f(x) = 7^{1-x} \) is a decreasing exponential function.
2. The function \( g(x) = 5^{2-3x} \) is also a decreasing exponential function.
The point of intersection is highlighted and labelled as (0.442, 2.964).
#### Task
Algebraically find the **EXACT** solution to \( f(x) = g(x) \) and then verify the three decimal approximation for the input value as the solution.
#### Solution
1. Set the functions equal to each other:
\[
7^{1-x} = 5^{2-3x}
\]
2. Solve for \( x \) by taking the natural logarithm of both sides.
#### Graph Explanation
- The black curve represents the function \( f(x) = 7^{1-x} \).
- The blue curve represents the function \( g(x) = 5^{2-3x} \).
- The intersection point at approximately \( x = 0.442 \) and \( y = 2.964 \) (marked in yellow) is where the two functions are equal.
By solving the equation \( 7^{1-x} = 5^{2-3x} \) analytically, students can find the exact value of \( x \) that satisfies the equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ecbc6b6-6518-454f-989a-d1b2a8923a43%2F3f81a62e-2ca5-46e0-8781-7862fb33d7e1%2Fu5wemg.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Intersection of Two Functions
#### Problem Statement
The functions given by \( f(x) = 7^{1-x} \) and \( g(x) = 5^{2-3x} \) are shown to intersect at a point as depicted in the figure below.
#### Graphical Representation
The graph presents the two functions \( f(x) = 7^{1-x} \) and \( g(x) = 5^{2-3x} \) plotted on a coordinate system. The x-axis ranges from 0 to 3, and the y-axis ranges from 0 to 10.
1. The function \( f(x) = 7^{1-x} \) is a decreasing exponential function.
2. The function \( g(x) = 5^{2-3x} \) is also a decreasing exponential function.
The point of intersection is highlighted and labelled as (0.442, 2.964).
#### Task
Algebraically find the **EXACT** solution to \( f(x) = g(x) \) and then verify the three decimal approximation for the input value as the solution.
#### Solution
1. Set the functions equal to each other:
\[
7^{1-x} = 5^{2-3x}
\]
2. Solve for \( x \) by taking the natural logarithm of both sides.
#### Graph Explanation
- The black curve represents the function \( f(x) = 7^{1-x} \).
- The blue curve represents the function \( g(x) = 5^{2-3x} \).
- The intersection point at approximately \( x = 0.442 \) and \( y = 2.964 \) (marked in yellow) is where the two functions are equal.
By solving the equation \( 7^{1-x} = 5^{2-3x} \) analytically, students can find the exact value of \( x \) that satisfies the equation.
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