Select The Correct Answer.What Is The $y$-intercept Of $f(x) = 3^{x+2}$?A. $(9,0)$ B. $(0,9)$ C. $(0,-9)$ D. $(9,-9)$

by ADMIN 133 views

The $y$-intercept of a function is a crucial concept in mathematics, particularly in algebra and calculus. It represents the point at which the graph of the function intersects the $y$-axis. In other words, it is the value of the function when the input, or the value of $x$, is equal to zero. In this article, we will explore the concept of the $y$-intercept and determine the correct answer for the given function $f(x) = 3^{x+2}$.

What is the $y$-Intercept?

The $y$-intercept is a point on the graph of a function where the value of $x$ is equal to zero. It is denoted by the symbol $y$ and is the value of the function when $x = 0$. The $y$-intercept is an essential concept in mathematics, as it provides valuable information about the behavior of a function.

Finding the $y$-Intercept of a Function

To find the $y$-intercept of a function, we need to substitute $x = 0$ into the function and solve for $y$. This will give us the value of the function when $x$ is equal to zero.

The Function $f(x) = 3^{x+2}$

The given function is $f(x) = 3^{x+2}$. To find the $y$-intercept of this function, we need to substitute $x = 0$ into the function and solve for $y$.

import math

def find_y_intercept():
x = 0
y = 3 ** (x + 2)
return y


y_intercept = find_y_intercept()
print(y_intercept)

Solving for $y$

When we substitute $x = 0$ into the function $f(x) = 3^{x+2}$, we get:

f(0)=30+2f(0) = 3^{0+2}

f(0)=32f(0) = 3^2

f(0)=9f(0) = 9

Therefore, the $y$-intercept of the function $f(x) = 3^{x+2}$ is $9$.

Conclusion

In conclusion, the $y$-intercept of a function is a point on the graph of the function where the value of $x$ is equal to zero. To find the $y$-intercept of a function, we need to substitute $x = 0$ into the function and solve for $y$. In this article, we found the $y$-intercept of the function $f(x) = 3^{x+2}$ to be $9$.

Answer

The correct answer is:

  • B. $(0,9)$

This is because the $y$-intercept of the function $f(x) = 3^{x+2}$ is $9$, which corresponds to the point $(0,9)$ on the graph of the function.

Discussion

The $y$-intercept of a function is an essential concept in mathematics, particularly in algebra and calculus. It provides valuable information about the behavior of a function and is used in a variety of applications, including physics, engineering, and economics. In this article, we explored the concept of the $y$-intercept and determined the correct answer for the given function $f(x) = 3^{x+2}$.

Related Topics

  • Graphing Functions: Graphing functions is an essential concept in mathematics, particularly in algebra and calculus. It involves plotting the graph of a function on a coordinate plane.
  • Solving Equations: Solving equations is a fundamental concept in mathematics, particularly in algebra and calculus. It involves finding the value of a variable that satisfies an equation.
  • Functions: Functions are a fundamental concept in mathematics, particularly in algebra and calculus. They involve a relationship between a set of inputs and a set of possible outputs.

References

  • Algebra and Calculus: Algebra and calculus are fundamental subjects in mathematics that deal with the study of functions, equations, and graphs.
  • Mathematics Textbooks: Mathematics textbooks provide a comprehensive overview of mathematical concepts, including functions, equations, and graphs.
  • Online Resources: Online resources, such as Khan Academy and MIT OpenCourseWare, provide a wealth of information on mathematical concepts, including functions, equations, and graphs.
    Q&A: Understanding the $y$-Intercept of a Function =====================================================

In our previous article, we explored the concept of the $y$-intercept of a function and determined the correct answer for the given function $f(x) = 3^{x+2}$. In this article, we will answer some frequently asked questions about the $y$-intercept of a function.

Q: What is the $y$-intercept of a function?

A: The $y$-intercept of a function is a point on the graph of the function where the value of $x$ is equal to zero. It is denoted by the symbol $y$ and is the value of the function when $x = 0$.

Q: How do I find the $y$-intercept of a function?

A: To find the $y$-intercept of a function, you need to substitute $x = 0$ into the function and solve for $y$. This will give you the value of the function when $x$ is equal to zero.

Q: What is the difference between the $y$-intercept and the $x$-intercept?

A: The $y$-intercept is the point on the graph of a function where the value of $x$ is equal to zero, while the $x$-intercept is the point on the graph of a function where the value of $y$ is equal to zero.

Q: Can the $y$-intercept be negative?

A: Yes, the $y$-intercept can be negative. For example, if the function is $f(x) = -3^{x+2}$, then the $y$-intercept would be $-9$.

Q: Can the $y$-intercept be a fraction?

A: Yes, the $y$-intercept can be a fraction. For example, if the function is $f(x) = \frac{3^{x+2}}{2}$, then the $y$-intercept would be $\frac{9}{2}$.

Q: How do I graph a function with a $y$-intercept?

A: To graph a function with a $y$-intercept, you need to plot the point on the graph of the function where the value of $x$ is equal to zero. This point is the $y$-intercept of the function.

Q: Can the $y$-intercept be a complex number?

A: Yes, the $y$-intercept can be a complex number. For example, if the function is $f(x) = 3^{x+2} + 4i$, then the $y$-intercept would be $9 + 4i$.

Q: How do I find the $y$-intercept of a function with a variable in the exponent?

A: To find the $y$-intercept of a function with a variable in the exponent, you need to substitute $x = 0$ into the function and solve for $y$. This will give you the value of the function when $x$ is equal to zero.

Q: Can the $y$-intercept be a function of another variable?

A: Yes, the $y$-intercept can be a function of another variable. For example, if the function is $f(x) = 3^{x+2} + g(x)$, then the $y$-intercept would be a function of $g(x)$.

Conclusion

In conclusion, the $y$-intercept of a function is a point on the graph of the function where the value of $x$ is equal to zero. It is denoted by the symbol $y$ and is the value of the function when $x = 0$. We hope that this article has helped to answer some of the frequently asked questions about the $y$-intercept of a function.

Related Topics

  • Graphing Functions: Graphing functions is an essential concept in mathematics, particularly in algebra and calculus. It involves plotting the graph of a function on a coordinate plane.
  • Solving Equations: Solving equations is a fundamental concept in mathematics, particularly in algebra and calculus. It involves finding the value of a variable that satisfies an equation.
  • Functions: Functions are a fundamental concept in mathematics, particularly in algebra and calculus. They involve a relationship between a set of inputs and a set of possible outputs.

References

  • Algebra and Calculus: Algebra and calculus are fundamental subjects in mathematics that deal with the study of functions, equations, and graphs.
  • Mathematics Textbooks: Mathematics textbooks provide a comprehensive overview of mathematical concepts, including functions, equations, and graphs.
  • Online Resources: Online resources, such as Khan Academy and MIT OpenCourseWare, provide a wealth of information on mathematical concepts, including functions, equations, and graphs.