### Graphical Analysis of FunctionsThe 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 ExplanationIn 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**

According to logarithm formula .

Answered: The graphs of f (x) = 2" and g(x) = 3"-…

Math

Algebra

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

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Concept explainers

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

### 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.

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