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 131 views

The $y$-intercept of a function is 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 $x$-value, is equal to zero. In this article, we will explore how to find the $y$-intercept of the function $f(x)=3^{x+2}$.

What is the $y$-Intercept?

The $y$-intercept is a fundamental concept in mathematics, particularly in algebra and calculus. It is used to determine the value of a function at a specific point, which can be useful in a variety of applications, such as modeling real-world phenomena or solving equations.

Finding the $y$-Intercept of $f(x)=3^{x+2}$

To find the $y$-intercept of the function $f(x)=3^{x+2}$, we need to substitute $x=0$ into the function and evaluate the result.

f(0) = 3^{0+2}
f(0) = 3^2
f(0) = 9

Therefore, the $y$-intercept of the function $f(x)=3^{x+2}$ is $(0,9)$.

Why is the $y$-Intercept Important?

The $y$-intercept is an important concept in mathematics because it provides a way to evaluate a function at a specific point. This can be useful in a variety of applications, such as:

  • Modeling real-world phenomena: The $y$-intercept can be used to model real-world phenomena, such as population growth or chemical reactions.
  • Solving equations: The $y$-intercept can be used to solve equations, such as quadratic equations or systems of equations.
  • Graphing functions: The $y$-intercept can be used to graph functions, which can be useful in visualizing complex relationships between variables.

Conclusion

In conclusion, the $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. To find the $y$-intercept of the function $f(x)=3^{x+2}$, we need to substitute $x=0$ into the function and evaluate the result. The $y$-intercept is an important concept in mathematics because it provides a way to evaluate a function at a specific point, which can be useful in a variety of applications.

Answer

The correct answer is:

  • B. $(0,9)$

Discussion

What is the $y$-intercept of the function $f(x)=3^{x+2}$? How do you find the $y$-intercept of a function? What are some applications of the $y$-intercept in mathematics?

Related Topics

  • Graphing functions: Graphing functions is an important concept in mathematics that involves visualizing complex relationships between variables.
  • Modeling real-world phenomena: Modeling real-world phenomena is an important application of mathematics that involves using mathematical models to describe and analyze real-world systems.
  • Solving equations: Solving equations is an important concept in mathematics that involves finding the value of a variable that satisfies a given equation.

References

  • Algebra: Algebra is a branch of mathematics that involves the study of variables and their relationships.
  • Calculus: Calculus is a branch of mathematics that involves the study of rates of change and accumulation.
  • Mathematics: Mathematics is a broad field of study that involves the study of numbers, quantities, and shapes.
    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 how to find it. In this article, we will answer some frequently asked questions about the $y$-intercept and provide additional examples to help solidify your understanding.

Q: What is the $y$-intercept of the function $f(x)=2x^2+3x-1$?

A: To find the $y$-intercept of the function $f(x)=2x^2+3x-1$, we need to substitute $x=0$ into the function and evaluate the result.

f(0) = 2(0)^2 + 3(0) - 1
f(0) = -1

Therefore, the $y$-intercept of the function $f(x)=2x^2+3x-1$ is $(0,-1)$.

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

A: To find the $y$-intercept of a rational function, we need to substitute $x=0$ into the function and evaluate the result. If the denominator is equal to zero, we need to simplify the function first.

For example, consider the rational function $f(x)=\frac{x+2}{x-1}$. To find the $y$-intercept, we need to substitute $x=0$ into the function and evaluate the result.

f(0) = \frac{0+2}{0-1}
f(0) = \frac{2}{-1}
f(0) = -2

Therefore, the $y$-intercept of the function $f(x)=\frac{x+2}{x-1}$ is $(0,-2)$.

Q: What is the $y$-intercept of the function $f(x)=\frac{1}{x}$?

A: To find the $y$-intercept of the function $f(x)=\frac{1}{x}$, we need to substitute $x=0$ into the function and evaluate the result. However, we cannot substitute $x=0$ into the function because it would result in an undefined value.

Instead, we can use the fact that the $y$-intercept is the value of the function when $x=0$. Since the function is undefined at $x=0$, we can say that the $y$-intercept of the function $f(x)=\frac{1}{x}$ is undefined.

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

A: To find the $y$-intercept of a function with a negative exponent, we need to substitute $x=0$ into the function and evaluate the result. If the exponent is negative, we need to take the reciprocal of the result.

For example, consider the function $f(x)=\frac{1}{x^2}$. To find the $y$-intercept, we need to substitute $x=0$ into the function and evaluate the result.

f(0) = \frac{1}{0^2}
f(0) = \frac{1}{0}
f(0) = \infty

Therefore, the $y$-intercept of the function $f(x)=\frac{1}{x^2}$ is undefined.

Q: What is the $y$-intercept of the function $f(x)=\sin(x)$?

A: To find the $y$-intercept of the function $f(x)=\sin(x)$, we need to substitute $x=0$ into the function and evaluate the result.

f(0) = \sin(0)
f(0) = 0

Therefore, the $y$-intercept of the function $f(x)=\sin(x)$ is $(0,0)$.

Conclusion

In conclusion, the $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. To find the $y$-intercept of a function, we need to substitute $x=0$ into the function and evaluate the result. We have answered some frequently asked questions about the $y$-intercept and provided additional examples to help solidify your understanding.

Answer

The correct answers are:

  • Q1: $(0,-1)$
  • Q2: $(0,-2)$
  • Q3: undefined
  • Q4: $(0,0)$

Discussion

What is the $y$-intercept of a function? How do you find the $y$-intercept of a function? What are some applications of the $y$-intercept in mathematics?

Related Topics

  • Graphing functions: Graphing functions is an important concept in mathematics that involves visualizing complex relationships between variables.
  • Modeling real-world phenomena: Modeling real-world phenomena is an important application of mathematics that involves using mathematical models to describe and analyze real-world systems.
  • Solving equations: Solving equations is an important concept in mathematics that involves finding the value of a variable that satisfies a given equation.

References

  • Algebra: Algebra is a branch of mathematics that involves the study of variables and their relationships.
  • Calculus: Calculus is a branch of mathematics that involves the study of rates of change and accumulation.
  • Mathematics: Mathematics is a broad field of study that involves the study of numbers, quantities, and shapes.