Consider The Graph Of The Function $f(x)=\log _2 X$.What Are The Features Of The Function $g$ If $g(x)=f(x+4)+8$?

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Introduction

When dealing with functions, transformations play a crucial role in understanding their behavior and characteristics. In this article, we will explore the transformation of the function g(x) based on the function f(x) = log2(x). We will analyze the features of the function g(x) = f(x+4) + 8 and discuss how it relates to the original function f(x).

The Original Function f(x)

The original function f(x) is defined as f(x) = log2(x). This function represents the logarithm of x with base 2. The graph of this function is a logarithmic curve that increases slowly as x increases.

The Transformation of g(x)

The function g(x) is defined as g(x) = f(x+4) + 8. To understand the transformation of g(x), we need to break it down into two parts: the horizontal shift and the vertical shift.

Horizontal Shift

The function f(x+4) represents a horizontal shift of the function f(x) by 4 units to the left. This means that the graph of f(x+4) is identical to the graph of f(x) but shifted 4 units to the left.

Vertical Shift

The function f(x+4) + 8 represents a vertical shift of the function f(x+4) by 8 units upwards. This means that the graph of f(x+4) + 8 is identical to the graph of f(x+4) but shifted 8 units upwards.

The Features of g(x)

Based on the transformation of g(x), we can identify the following features:

  • Domain: The domain of g(x) is the same as the domain of f(x), which is all positive real numbers.
  • Range: The range of g(x) is the same as the range of f(x) + 8, which is all real numbers.
  • Asymptote: The asymptote of g(x) is the same as the asymptote of f(x), which is the y-axis.
  • Intercepts: The x-intercept of g(x) is the same as the x-intercept of f(x+4), which is 2^(-4) = 1/16. The y-intercept of g(x) is 8.
  • Increasing/Decreasing: The function g(x) is increasing for all x > 0.

Graphical Representation

The graph of g(x) can be represented as a logarithmic curve that increases slowly as x increases. The graph is shifted 4 units to the left and 8 units upwards compared to the graph of f(x).

Conclusion

In conclusion, the function g(x) = f(x+4) + 8 is a transformation of the function f(x) = log2(x). The transformation involves a horizontal shift of 4 units to the left and a vertical shift of 8 units upwards. The features of g(x) include a domain of all positive real numbers, a range of all real numbers, an asymptote of the y-axis, x-intercept of 1/16, and a y-intercept of 8. The graph of g(x) is a logarithmic curve that increases slowly as x increases.

Key Takeaways

  • The transformation of g(x) involves a horizontal shift of 4 units to the left and a vertical shift of 8 units upwards.
  • The features of g(x) include a domain of all positive real numbers, a range of all real numbers, an asymptote of the y-axis, x-intercept of 1/16, and a y-intercept of 8.
  • The graph of g(x) is a logarithmic curve that increases slowly as x increases.

Further Reading

For further reading on the topic of function transformations, we recommend the following resources:

  • [1] Khan Academy: Function Transformations
  • [2] Math Open Reference: Function Transformations
  • [3] Wolfram MathWorld: Function Transformations

References

[1] Khan Academy. (n.d.). Function Transformations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f1f1/x2f1f1f1-function-transformations

[2] Math Open Reference. (n.d.). Function Transformations. Retrieved from https://www.mathopenref.com/functiontransformations.html

[3] Wolfram MathWorld. (n.d.). Function Transformations. Retrieved from https://mathworld.wolfram.com/FunctionTransformations.html

Introduction

In our previous article, we explored the transformation of the function g(x) based on the function f(x) = log2(x). We analyzed the features of the function g(x) = f(x+4) + 8 and discussed how it relates to the original function f(x). In this article, we will answer some frequently asked questions about the transformation of the function g(x).

Q1: What is the domain of the function g(x)?

A1: The domain of the function g(x) is the same as the domain of f(x), which is all positive real numbers.

Q2: What is the range of the function g(x)?

A2: The range of the function g(x) is the same as the range of f(x) + 8, which is all real numbers.

Q3: What is the asymptote of the function g(x)?

A3: The asymptote of the function g(x) is the same as the asymptote of f(x), which is the y-axis.

Q4: What is the x-intercept of the function g(x)?

A4: The x-intercept of the function g(x) is the same as the x-intercept of f(x+4), which is 2^(-4) = 1/16.

Q5: What is the y-intercept of the function g(x)?

A5: The y-intercept of the function g(x) is 8.

Q6: Is the function g(x) increasing or decreasing?

A6: The function g(x) is increasing for all x > 0.

Q7: How does the transformation of g(x) affect the graph of f(x)?

A7: The transformation of g(x) involves a horizontal shift of 4 units to the left and a vertical shift of 8 units upwards. This means that the graph of g(x) is identical to the graph of f(x) but shifted 4 units to the left and 8 units upwards.

Q8: What is the significance of the horizontal shift in the transformation of g(x)?

A8: The horizontal shift of 4 units to the left means that the graph of g(x) is shifted 4 units to the left compared to the graph of f(x).

Q9: What is the significance of the vertical shift in the transformation of g(x)?

A9: The vertical shift of 8 units upwards means that the graph of g(x) is shifted 8 units upwards compared to the graph of f(x+4).

Q10: How can I visualize the transformation of g(x)?

A10: You can visualize the transformation of g(x) by plotting the graph of f(x) and then shifting it 4 units to the left and 8 units upwards to obtain the graph of g(x).

Conclusion

In conclusion, the transformation of the function g(x) involves a horizontal shift of 4 units to the left and a vertical shift of 8 units upwards. The features of g(x) include a domain of all positive real numbers, a range of all real numbers, an asymptote of the y-axis, x-intercept of 1/16, and a y-intercept of 8. The graph of g(x) is a logarithmic curve that increases slowly as x increases.

Key Takeaways

  • The transformation of g(x) involves a horizontal shift of 4 units to the left and a vertical shift of 8 units upwards.
  • The features of g(x) include a domain of all positive real numbers, a range of all real numbers, an asymptote of the y-axis, x-intercept of 1/16, and a y-intercept of 8.
  • The graph of g(x) is a logarithmic curve that increases slowly as x increases.

Further Reading

For further reading on the topic of function transformations, we recommend the following resources:

  • [1] Khan Academy: Function Transformations
  • [2] Math Open Reference: Function Transformations
  • [3] Wolfram MathWorld: Function Transformations

References

[1] Khan Academy. (n.d.). Function Transformations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f1f1/x2f1f1f1-function-transformations

[2] Math Open Reference. (n.d.). Function Transformations. Retrieved from https://www.mathopenref.com/functiontransformations.html

[3] Wolfram MathWorld. (n.d.). Function Transformations. Retrieved from https://mathworld.wolfram.com/FunctionTransformations.html