Problem 131
Question
Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$3^{x}=2 x+3$$
Step-by-Step Solution
Verified Answer
The \(x\)-coordinate of the intersection point of the plots of \(y = 3^{x}\) and \(y = 2x + 3\) is the solution, which should then be verified by substitution into the original equation.
1Step 1 - Graph the exponential function
Graph the function \(y = 3^{x}\). This is an exponential function with base 3.
2Step 2 - Graph the linear function
Next, graph the function \(y = 2x + 3\) on the same set of axes as the exponential function. This is a linear function with a slope of 2 and a y-intercept of 3.
3Step 3 - Find the intersection point
Identify the point where the two functions intersect. The x-coordinate of this point is the solution to the equation.
4Step 4 - Verify the solution
Substitute the value of the obtained x in the original equation and check that both sides of the equation are equal. If they are, then the solution is verified.
Key Concepts
Exponential FunctionLinear FunctionIntersection PointVerifying Solutions
Exponential Function
An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In this case, the function is given as \(y = 3^x\). Its key characteristic is its rapid growth as the value of \(x\) increases.
This rapid increase makes exponential functions powerful tools for modeling real-world situations such as population growth or radioactive decay.
To graph \(3^x\), plot some points by choosing values of \(x\), calculate the corresponding \(y\) values, and then plot these on the coordinate plane.
As you do this, you'll notice the steep upward curve. This curve starts nearly flat and rises steeply as \(x\) becomes larger, reflecting the function's exponential nature.
This rapid increase makes exponential functions powerful tools for modeling real-world situations such as population growth or radioactive decay.
To graph \(3^x\), plot some points by choosing values of \(x\), calculate the corresponding \(y\) values, and then plot these on the coordinate plane.
As you do this, you'll notice the steep upward curve. This curve starts nearly flat and rises steeply as \(x\) becomes larger, reflecting the function's exponential nature.
Linear Function
A linear function is a straight-line graph. It's often in the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
In this exercise, the linear function is \(y = 2x + 3\). Here, the slope \(m = 2\) tells us the line rises by 2 units for each unit it moves to the right. Meanwhile, the y-intercept \(b = 3\) is the point where the line crosses the y-axis.
This consistent rate of increase creates a straight line, making linear functions useful for modeling constant rates, like speed or market trends.
To graph it, start at the y-intercept (0, 3), and use the slope to find more points by moving up 2 units for every 1 unit to the right across the coordinate plane.
In this exercise, the linear function is \(y = 2x + 3\). Here, the slope \(m = 2\) tells us the line rises by 2 units for each unit it moves to the right. Meanwhile, the y-intercept \(b = 3\) is the point where the line crosses the y-axis.
This consistent rate of increase creates a straight line, making linear functions useful for modeling constant rates, like speed or market trends.
To graph it, start at the y-intercept (0, 3), and use the slope to find more points by moving up 2 units for every 1 unit to the right across the coordinate plane.
Intersection Point
The intersection point of two functions on a graph is crucial as it represents a solution to the equation that sets them equal. It's where the two different functions, here \(y = 3^x\) and \(y = 2x + 3\), intersect on the graph.
Finding this point visually on a graph helps to identify the \(x\) value that satisfies the equation \(3^x = 2x + 3\).
This point signifies the exact input where both functions yield the same output value, thereby solving the equation.
Use graphing tools or precise plotting to locate the intersection accurately. Knowing this supports understanding of how the exponential and linear graphs relate to each other.
Finding this point visually on a graph helps to identify the \(x\) value that satisfies the equation \(3^x = 2x + 3\).
This point signifies the exact input where both functions yield the same output value, thereby solving the equation.
Use graphing tools or precise plotting to locate the intersection accurately. Knowing this supports understanding of how the exponential and linear graphs relate to each other.
Verifying Solutions
After determining the intersection point, verifying your solution is essential to ensure its accuracy.
To verify, substitute the \(x\)-value from the intersection point back into the original equation \(3^x = 2x + 3\) to check if both sides equal.
Here, substitute the \(x\)-coordinate, let's call it \(x_0\), into both the left-hand side and right-hand side of the equation.
To verify, substitute the \(x\)-value from the intersection point back into the original equation \(3^x = 2x + 3\) to check if both sides equal.
Here, substitute the \(x\)-coordinate, let's call it \(x_0\), into both the left-hand side and right-hand side of the equation.
- If \(3^{x_0} = 2(x_0) + 3\), then the solution is correct, confirming the point of intersection is valid.
- If not, check calculations or graphing methods for mistakes.
Other exercises in this chapter
Problem 130
Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\log x, g(x)=-\log x$$
View solution Problem 130
If \(\log 3=A\) and \(\log 7=B,\) find \(\log _{7} 9\) in terms of \(A\) and \(B\).
View solution Problem 131
Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\log x, g(x)=\log (x-2)+1$$
View solution Problem 131
Write as a single term that does not contain a logarithm: $$e^{\ln 8 x^{5}-\ln 2 x^{2}}$$
View solution