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The graphs of f (x) = 2" and g(x) = 3"- are shown. %3D

The graphs of f (x) = 2" and g(x) = 3"- are shown. %3D

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
ChapterP: Prerequisites
SectionP.7: A Library Of Parent Functions
Problem 47E
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### Graphical Analysis of Functions The graphs of the functions \( f(x) = 2^x \) and \( g(x) = 3^x - 1 \) are shown below. #### Function Descriptions - **\( f(x) = 2^x \)**: This represents an exponential function where the base is 2. The graph of this function is a curve that starts close to the x-axis (for negative values of x), increases steadily as x approaches zero, and then rises sharply for positive values of x. - **\( g(x) = 3^x - 1 \)**: This function is another exponential function where the base is 3, but it is shifted downward by 1 unit. The graph follows a similar pattern to \( f(x) \) but starts at a different location due to the vertical shift. #### Graph Explanation In the provided graph: - The horizontal axis (x-axis) and the vertical axis (y-axis) intersect at the origin (0,0). - The curve of \( f(x) = 2^x \) starts near the x-axis for negative x values, and rises exponentially as x becomes positive. - The curve of \( g(x) = 3^x - 1 \) also starts near the x-axis for negative x values but slightly below the curve of \( f(x) \) because it has been shifted down by 1 unit. It rises more steeply than the graph of \( f(x) \) because the base (3) is larger than 2. - As x increases, both functions exhibit exponential growth, but \( g(x) \) increases faster than \( f(x) \) due to its higher base. Understanding these exponential functions and their graphical representations is crucial in various fields including mathematics, science, and engineering, as they often model real-world phenomena such as population growth, radioactive decay, and financial investments.
Transcribed Image Text:### Graphical Analysis of Functions The graphs of the functions \( f(x) = 2^x \) and \( g(x) = 3^x - 1 \) are shown below. #### Function Descriptions - **\( f(x) = 2^x \)**: This represents an exponential function where the base is 2. The graph of this function is a curve that starts close to the x-axis (for negative values of x), increases steadily as x approaches zero, and then rises sharply for positive values of x. - **\( g(x) = 3^x - 1 \)**: This function is another exponential function where the base is 3, but it is shifted downward by 1 unit. The graph follows a similar pattern to \( f(x) \) but starts at a different location due to the vertical shift. #### Graph Explanation In the provided graph: - The horizontal axis (x-axis) and the vertical axis (y-axis) intersect at the origin (0,0). - The curve of \( f(x) = 2^x \) starts near the x-axis for negative x values, and rises exponentially as x becomes positive. - The curve of \( g(x) = 3^x - 1 \) also starts near the x-axis for negative x values but slightly below the curve of \( f(x) \) because it has been shifted down by 1 unit. It rises more steeply than the graph of \( f(x) \) because the base (3) is larger than 2. - As x increases, both functions exhibit exponential growth, but \( g(x) \) increases faster than \( f(x) \) due to its higher base. Understanding these exponential functions and their graphical representations is crucial in various fields including mathematics, science, and engineering, as they often model real-world phenomena such as population growth, radioactive decay, and financial investments.
### Solving Equations Graphically and Algebraically **Graphical Analysis:** The graph above depicts two functions, \( f(x) \) and \( g(x) \). The solution to the equation \( f(x) = g(x) \) is the \( x \)-coordinate of the point where the two lines intersect. **Steps to Identify the Solution Graphically:** 1. Observe the graph to locate the point of intersection. 2. Identify the \( x \)-coordinate of this point. #### Example Task: **a.** The solution to the equation \( f(x) = g(x) \) is the \( \boxed{} \) coordinate of the point where the two lines intersect. - The solution is approximately \( \boxed{} \). **Algebraic Solution:** **b.** Solve the equation \( 2^x = 3^{x-1} \) algebraically. Round your answer to 4 decimal places. **Steps for Solving Algebraically:** 1. Set the equations equal: \( 2^x = 3^{x-1} \). 2. Apply logarithms on both sides to solve for \( x \). 3. Round the final solution to 4 decimal places. \( \boxed{} \) **Note:** The provided fields in boxes allow students to input their solutions as they follow the instructions. --- **Privacy Policy** | **Credits** | **CA Residents: Do Not Sell My Info**
Transcribed Image Text:### Solving Equations Graphically and Algebraically **Graphical Analysis:** The graph above depicts two functions, \( f(x) \) and \( g(x) \). The solution to the equation \( f(x) = g(x) \) is the \( x \)-coordinate of the point where the two lines intersect. **Steps to Identify the Solution Graphically:** 1. Observe the graph to locate the point of intersection. 2. Identify the \( x \)-coordinate of this point. #### Example Task: **a.** The solution to the equation \( f(x) = g(x) \) is the \( \boxed{} \) coordinate of the point where the two lines intersect. - The solution is approximately \( \boxed{} \). **Algebraic Solution:** **b.** Solve the equation \( 2^x = 3^{x-1} \) algebraically. Round your answer to 4 decimal places. **Steps for Solving Algebraically:** 1. Set the equations equal: \( 2^x = 3^{x-1} \). 2. Apply logarithms on both sides to solve for \( x \). 3. Round the final solution to 4 decimal places. \( \boxed{} \) **Note:** The provided fields in boxes allow students to input their solutions as they follow the instructions. --- **Privacy Policy** | **Credits** | **CA Residents: Do Not Sell My Info**
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