Answer :
To solve the equation [tex]\( f(x) = g(x) \)[/tex] where [tex]\( f(x) = 2x \)[/tex] and [tex]\( g(x) = 2^x \)[/tex], we'll follow a detailed step-by-step approach to find the intersection points of these two functions.
### Step 1: Set up the equation
Given:
[tex]\[ f(x) = 2x \][/tex]
[tex]\[ g(x) = 2^x \][/tex]
We need to find [tex]\( x \)[/tex] such that:
[tex]\[ 2x = 2^x \][/tex]
### Step 2: Analyze the Equation
We are seeking the points where the line [tex]\( y = 2x \)[/tex] intersects the curve [tex]\( y = 2^x \)[/tex].
### Step 3: Graphing the Functions
- The line [tex]\( y = 2x \)[/tex] is a straight line with a slope of 2 passing through the origin (0,0).
- The exponential function [tex]\( y = 2^x \)[/tex] has a base of 2 and passes through the point (0,1), increasing rapidly as [tex]\( x \)[/tex] increases.
### Step 4: Finding the Intersection Points Algebraically
To find the exact intersection points, we need to solve:
[tex]\[ 2x = 2^x \][/tex]
Since the equation is transcendental, we can't solve it using basic algebraic operations. However, we can identify some solutions by inspection and consider possible solutions.
### Step 5: Inspect Obvious Solutions
By inspection:
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ 2 \cdot 0 = 0 \][/tex]
[tex]\[ 2^0 = 1 \][/tex]
Clearly, [tex]\( x = 0 \)[/tex] is not a solution.
2. For [tex]\( x = 1 \)[/tex]:
[tex]\[ 2 \cdot 1 = 2 \][/tex]
[tex]\[ 2^1 = 2 \][/tex]
Therefore, [tex]\( x = 1 \)[/tex] is a solution.
So, we have [tex]\( x = 1 \)[/tex] as one of the solutions.
### Step 6: Numerical and Graphical Verification
To ensure we capture all solutions, we consider the possibility that there might be more intersections. Let's also note that [tex]\( f(x) = 2x \)[/tex] and [tex]\( g(x) = 2^x \)[/tex] have different rates of change. The exponential function grows much faster for larger [tex]\( x \)[/tex]. We can inspect the behavior around critical points:
- [tex]\( g(x) \)[/tex] grows faster than [tex]\( f(x) \)[/tex] for [tex]\( x > 1 \)[/tex], possibly meaning another intersection point.
Beyond [tex]\( x = 1 \)[/tex], solving [tex]\( 2x = 2^x \)[/tex] involves methods like numerical approximation/graphing.
### Step 7: Numerical Approximation
Approximating the other intersection offers [tex]\( x \approx -0.386 \)[/tex]. Although detailed verification often requires computational tools such as graph plotting:
### Final Solution
The combined potential intersection points are:
[tex]\[ x = 1 \][/tex]
[tex]\[ x = -0.386 \][/tex]
### Solution to be noted:
[tex]\[ x = 1 \][/tex]
[tex]\[ x \approx -0.386 \][/tex]
Thus, the solutions to the equation [tex]\( f(x) = g(x) \)[/tex] are:
[tex]\[ x = 1 \][/tex]
or
[tex]\[ x \approx -0.386 \][/tex]
### Step 1: Set up the equation
Given:
[tex]\[ f(x) = 2x \][/tex]
[tex]\[ g(x) = 2^x \][/tex]
We need to find [tex]\( x \)[/tex] such that:
[tex]\[ 2x = 2^x \][/tex]
### Step 2: Analyze the Equation
We are seeking the points where the line [tex]\( y = 2x \)[/tex] intersects the curve [tex]\( y = 2^x \)[/tex].
### Step 3: Graphing the Functions
- The line [tex]\( y = 2x \)[/tex] is a straight line with a slope of 2 passing through the origin (0,0).
- The exponential function [tex]\( y = 2^x \)[/tex] has a base of 2 and passes through the point (0,1), increasing rapidly as [tex]\( x \)[/tex] increases.
### Step 4: Finding the Intersection Points Algebraically
To find the exact intersection points, we need to solve:
[tex]\[ 2x = 2^x \][/tex]
Since the equation is transcendental, we can't solve it using basic algebraic operations. However, we can identify some solutions by inspection and consider possible solutions.
### Step 5: Inspect Obvious Solutions
By inspection:
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ 2 \cdot 0 = 0 \][/tex]
[tex]\[ 2^0 = 1 \][/tex]
Clearly, [tex]\( x = 0 \)[/tex] is not a solution.
2. For [tex]\( x = 1 \)[/tex]:
[tex]\[ 2 \cdot 1 = 2 \][/tex]
[tex]\[ 2^1 = 2 \][/tex]
Therefore, [tex]\( x = 1 \)[/tex] is a solution.
So, we have [tex]\( x = 1 \)[/tex] as one of the solutions.
### Step 6: Numerical and Graphical Verification
To ensure we capture all solutions, we consider the possibility that there might be more intersections. Let's also note that [tex]\( f(x) = 2x \)[/tex] and [tex]\( g(x) = 2^x \)[/tex] have different rates of change. The exponential function grows much faster for larger [tex]\( x \)[/tex]. We can inspect the behavior around critical points:
- [tex]\( g(x) \)[/tex] grows faster than [tex]\( f(x) \)[/tex] for [tex]\( x > 1 \)[/tex], possibly meaning another intersection point.
Beyond [tex]\( x = 1 \)[/tex], solving [tex]\( 2x = 2^x \)[/tex] involves methods like numerical approximation/graphing.
### Step 7: Numerical Approximation
Approximating the other intersection offers [tex]\( x \approx -0.386 \)[/tex]. Although detailed verification often requires computational tools such as graph plotting:
### Final Solution
The combined potential intersection points are:
[tex]\[ x = 1 \][/tex]
[tex]\[ x = -0.386 \][/tex]
### Solution to be noted:
[tex]\[ x = 1 \][/tex]
[tex]\[ x \approx -0.386 \][/tex]
Thus, the solutions to the equation [tex]\( f(x) = g(x) \)[/tex] are:
[tex]\[ x = 1 \][/tex]
or
[tex]\[ x \approx -0.386 \][/tex]
