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Introduction

In mathematics, functions are essential tools for modeling real-world phenomena. When dealing with functions, it's crucial to understand their key features, such as domain, range, and behavior. In this article, we'll explore the transformation of functions using the graph of the function $f(x) = 2^x$ as a reference. We'll examine the statements that describe key features of the function $g(x) = f(x+2)$.

The Original Function: $f(x) = 2^x$

The function $f(x) = 2^x$ is an exponential function with a base of 2. Its graph is a curve that increases rapidly as $x$ increases. The domain of $f(x)$ is all real numbers, denoted as $(-\infty, \infty)$.

The Transformed Function: $g(x) = f(x+2)$

To obtain the function $g(x)$, we replace $x$ with $x+2$ in the original function $f(x)$. This means that the graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ by 2 units to the left.

Key Features of $g(x)$

Now, let's examine the statements that describe key features of the function $g(x)$.

Domain of $g(x)$

The domain of $g(x)$ is the set of all possible input values for which the function is defined. Since the graph of $g(x)$ is a horizontal shift of the graph of $f(x)$, the domain of $g(x)$ is also $(-\infty, \infty)$.

Range of $g(x)$

The range of $g(x)$ is the set of all possible output values for which the function is defined. Since the graph of $g(x)$ is an exponential function, its range is all positive real numbers, denoted as $(0, \infty)$.

Horizontal Shift

As mentioned earlier, the graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ by 2 units to the left. This means that for any point $(x, y)$ on the graph of $f(x)$, the corresponding point on the graph of $g(x)$ is $(x-2, y)$.

Vertical Stretch

The graph of $g(x)$ is also a vertical stretch of the graph of $f(x)$. This means that for any point $(x, y)$ on the graph of $f(x)$, the corresponding point on the graph of $g(x)$ is $(x, 2y)$.

Asymptotes

The graph of $g(x)$ has two asymptotes: the x-axis and the y-axis. The x-axis is a horizontal asymptote, and the y-axis is a vertical asymptote.

Intercepts

The graph of $g(x)$ has two x-intercepts: $(-2, 0)$ and $(0, 0)$. The graph also has a y-intercept at $(0, 1)$.

Increasing/Decreasing Intervals

The graph of $g(x)$ is increasing for all $x$ in its domain. This means that the function is always increasing, and there are no intervals where the function is decreasing.

Local Maxima/Minima

The graph of $g(x)$ has no local maxima or minima. This means that the function is always increasing, and there are no points on the graph where the function changes from increasing to decreasing or vice versa.

Inflection Points

The graph of $g(x)$ has no inflection points. This means that the function is always concave up, and there are no points on the graph where the function changes from concave up to concave down or vice versa.

Conclusion

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Introduction

In our previous article, we explored the transformation of functions using the graph of the function $f(x) = 2^x$ as a reference. We examined the statements that describe key features of the function $g(x) = f(x+2)$. In this article, we'll answer some frequently asked questions about the transformation of functions.

Q: What is the effect of replacing $x$ with $x+2$ in the function $f(x)$?

A: Replacing $x$ with $x+2$ in the function $f(x)$ shifts the graph of $f(x)$ to the left by 2 units. This means that for any point $(x, y)$ on the graph of $f(x)$, the corresponding point on the graph of $g(x)$ is $(x-2, y)$.

Q: How does the domain of $g(x)$ compare to the domain of $f(x)$?

A: The domain of $g(x)$ is the same as the domain of $f(x)$, which is $(-\infty, \infty)$. This means that $g(x)$ is defined for all real numbers.

Q: What is the effect of the horizontal shift on the range of $g(x)$?

A: The horizontal shift does not affect the range of $g(x)$. The range of $g(x)$ is still $(0, \infty)$, which means that $g(x)$ is always positive.

Q: How does the vertical stretch affect the graph of $g(x)$?

A: The vertical stretch affects the graph of $g(x)$ by multiplying the y-coordinates of the points on the graph by 2. This means that for any point $(x, y)$ on the graph of $f(x)$, the corresponding point on the graph of $g(x)$ is $(x, 2y)$.

Q: What are the asymptotes of the graph of $g(x)$?

A: The graph of $g(x)$ has two asymptotes: the x-axis and the y-axis. The x-axis is a horizontal asymptote, and the y-axis is a vertical asymptote.

Q: How many x-intercepts does the graph of $g(x)$ have?

A: The graph of $g(x)$ has two x-intercepts: $(-2, 0)$ and $(0, 0)$.

Q: What is the y-intercept of the graph of $g(x)$?

A: The y-intercept of the graph of $g(x)$ is $(0, 1)$.

Q: Is the graph of $g(x)$ always increasing or decreasing?

A: The graph of $g(x)$ is always increasing. There are no intervals where the function is decreasing.

Q: Does the graph of $g(x)$ have any local maxima or minima?

A: No, the graph of $g(x)$ does not have any local maxima or minima. The function is always increasing.

Q: Does the graph of $g(x)$ have any inflection points?

A: No, the graph of $g(x)$ does not have any inflection points. The function is always concave up.

Conclusion

In conclusion, the transformation of functions is an essential concept in mathematics. By understanding how functions are transformed, we can analyze and solve problems involving functions. In this article, we answered some frequently asked questions about the transformation of functions, including the effect of replacing $x$ with $x+2$, the domain and range of $g(x)$, the asymptotes, x-intercepts, y-intercept, and local maxima or minima.

Additional Resources

For more information on the transformation of functions, we recommend the following resources:

• Khan Academy: Functions and Graphs
• Mathway: Function Graphing
• Wolfram Alpha: Function Transformation

We hope this article has been helpful in understanding the transformation of functions. If you have any further questions or need additional clarification, please don't hesitate to ask.