Function { G $}$ Is A Transformation Of The Parent Exponential Function. Which Statements Are True About Function { G $}$?1. Function { G $}$ Has A { Y $}$-intercept Of { (0, 4)$}$.2. The Range Of

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Introduction

In mathematics, transformations of functions are essential concepts that help us understand how different functions are related to each other. One such transformation is the function { g $}$, which is a transformation of the parent exponential function. In this article, we will explore the properties of function { g $}$ and determine which statements are true about it.

The Parent Exponential Function

The parent exponential function is given by the equation { f(x) = b^x $}$, where { b $}$ is a positive constant. This function has a { y $}$-intercept of {(0, 1)$}$ and a range of {(0, \infty)$}$. The graph of the parent exponential function is a curve that increases rapidly as { x $}$ increases.

Transformation of the Parent Exponential Function

The function { g $}$ is a transformation of the parent exponential function. This means that { g $}$ is a function that is obtained by applying one or more transformations to the parent exponential function. The general form of { g $}$ is given by { g(x) = a \cdot f(bx + c) + d $}$, where { a $}$, { b $}$, { c $}$, and { d $}$ are constants.

Statement 1: Function { g $}$ has a { y $}$-intercept of {(0, 4)$}$

To determine if this statement is true, we need to find the { y $}$-intercept of { g $}$. The { y $}$-intercept of a function is the point where the graph of the function intersects the { y $}$-axis. In other words, it is the value of { y $}$ when { x $}$ is equal to zero.

Let's assume that the { y $}$-intercept of { g $}$ is {(0, 4)$}$. This means that when { x $}$ is equal to zero, the value of { g(x) $}$ is equal to four.

We can write this as an equation: { g(0) = 4 $}$. Substituting the general form of { g $}$ into this equation, we get:

{ a \cdot f(b \cdot 0 + c) + d = 4 $}$

Simplifying this equation, we get:

{ a \cdot f(c) + d = 4 $}$

Since { f(c) $}$ is equal to { b^c $}$, we can substitute this into the equation:

{ a \cdot b^c + d = 4 $}$

Now, we need to find the values of { a $}$, { b $}$, { c $}$, and { d $}$ that satisfy this equation.

Unfortunately, there is no unique solution to this equation. The values of { a $}$, { b $}$, { c $}$, and { d $}$ can be chosen arbitrarily, and the equation will still be satisfied.

Therefore, we cannot conclude that the { y $}$-intercept of { g $}$ is {(0, 4)$}$. This statement is false.

Statement 2: The range of { g $}$ is {(0, \infty)$)

To determine if this statement is true, we need to find the range of [$ g $}$. The range of a function is the set of all possible values of the function.

Since { g $}$ is a transformation of the parent exponential function, we know that the range of { g $}$ is also {(0, \infty)$}$. This is because the parent exponential function has a range of {(0, \infty)$}$, and the transformation does not change the range.

Therefore, this statement is true.

Conclusion

In conclusion, we have analyzed the properties of function { g $}$ and determined which statements are true about it. We found that statement 1 is false, and statement 2 is true.

Key Takeaways

  • The function { g $}$ is a transformation of the parent exponential function.
  • The range of { g $}$ is {(0, \infty)$}$.
  • The { y $}$-intercept of { g $}$ is not necessarily {(0, 4)$}$.

Further Reading

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

  • Khan Academy: Transformations of Functions
  • Math Is Fun: Transformations of Functions
  • Wolfram MathWorld: Transformations of Functions

References

  • [1] Larson, R. (2015). Calculus. Cengage Learning.
  • [2] Rogawski, J. (2015). Calculus: Early Transcendentals. W.H. Freeman and Company.
  • [3] Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
    Q&A: Function { g $}$ and Its Transformations =====================================================

Introduction

In our previous article, we explored the properties of function { g $}$ and determined which statements are true about it. In this article, we will answer some frequently asked questions about function { g $}$ and its transformations.

Q: What is the general form of function { g $}$?

A: The general form of function { g $}$ is given by { g(x) = a \cdot f(bx + c) + d $}$, where { a $}$, { b $}$, { c $}$, and { d $}$ are constants.

Q: What is the parent exponential function?

A: The parent exponential function is given by the equation { f(x) = b^x $}$, where { b $}$ is a positive constant. This function has a { y $}$-intercept of {(0, 1)$}$ and a range of {(0, \infty)$}$.

Q: What is the range of function { g $}$?

A: The range of function { g $}$ is {(0, \infty)$}$. This is because the parent exponential function has a range of {(0, \infty)$}$, and the transformation does not change the range.

Q: Is the { y $}$-intercept of function { g $}$ always {(0, 4)$]?

A: No, the [$ y $}$-intercept of function { g $}$ is not always {(0, 4)$}$. The { y $}$-intercept of { g $}$ depends on the values of the constants { a $}$, { b $}$, { c $}$, and { d $}$.

Q: How do I find the values of the constants { a $}$, { b $}$, { c $}$, and { d $}$?

A: To find the values of the constants { a $}$, { b $}$, { c $}$, and { d $}$, you need to use the given information about the function { g $}$. For example, if you know the { y $}$-intercept of { g $}$, you can use it to find the value of { d $}$.

Q: What are some common transformations of the parent exponential function?

A: Some common transformations of the parent exponential function include:

  • Vertical stretch or compression: { g(x) = a \cdot f(x) $}$
  • Horizontal stretch or compression: { g(x) = f(bx) $}$
  • Reflection across the { x $}$-axis: { g(x) = -f(x) $}$
  • Reflection across the { y $}$-axis: { g(x) = f(-x) $}$

Q: How do I graph the function { g $}$?

A: To graph the function { g $}$, you need to use the general form of the function and the given information about the constants { a $}$, { b $}$, { c $}$, and { d $}$. You can use a graphing calculator or software to graph the function.

Conclusion

In conclusion, we have answered some frequently asked questions about function { g $}$ and its transformations. We hope this article has been helpful in understanding the properties of function { g $}$ and its transformations.

Key Takeaways

  • The general form of function { g $}$ is given by { g(x) = a \cdot f(bx + c) + d $}$.
  • The range of function { g $}$ is {(0, \infty)$}$.
  • The { y $}$-intercept of function { g $}$ depends on the values of the constants { a $}$, { b $}$, { c $}$, and { d $}$.

Further Reading

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

  • Khan Academy: Transformations of Functions
  • Math Is Fun: Transformations of Functions
  • Wolfram MathWorld: Transformations of Functions

References

  • [1] Larson, R. (2015). Calculus. Cengage Learning.
  • [2] Rogawski, J. (2015). Calculus: Early Transcendentals. W.H. Freeman and Company.
  • [3] Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.