What Is The $y$-intercept Of The Function $f(x) = 4 - 5x$?A. − 5 -5 − 5 B. − 4 -4 − 4 C. 4 4 4 D. 5 5 5

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Understanding the $y$-intercept

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 $y$ when $x$ is equal to zero. To find the $y$-intercept of a linear function, we can substitute $x = 0$ into the equation of the function.

Finding the $y$-intercept of the function $f(x) = 4 - 5x$

To find the $y$-intercept of the function $f(x) = 4 - 5x$, we can substitute $x = 0$ into the equation of the function.

f(x)=4−5xf(x) = 4 - 5x

f(0)=4−5(0)f(0) = 4 - 5(0)

f(0)=4−0f(0) = 4 - 0

f(0)=4f(0) = 4

Therefore, the $y$-intercept of the function $f(x) = 4 - 5x$ is $4$.

Conclusion

The $y$-intercept of a function is an important concept in mathematics, particularly in algebra and calculus. It is the point at which the graph of the function intersects the $y$-axis, and it can be used to determine the value of the function at a specific point. In this article, we have discussed how to find the $y$-intercept of a linear function, and we have used the function $f(x) = 4 - 5x$ as an example. We have shown that the $y$-intercept of this function is $4$.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

  1. Understand the concept of the $y$-intercept: The $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis.
  2. Substitute $x = 0$ into the equation of the function: To find the $y$-intercept of a linear function, we can substitute $x = 0$ into the equation of the function.
  3. Simplify the equation: After substituting $x = 0$ into the equation of the function, we can simplify the equation to find the value of the function at $x = 0$.
  4. Determine the $y$-intercept: The value of the function at $x = 0$ is the $y$-intercept of the function.

Common Mistakes

Here are some common mistakes to avoid when finding the $y$-intercept of a function:

  • Not substituting $x = 0$ into the equation of the function: This is the most common mistake when finding the $y$-intercept of a function.
  • Not simplifying the equation: After substituting $x = 0$ into the equation of the function, we need to simplify the equation to find the value of the function at $x = 0$.
  • Not determining the $y$-intercept: After simplifying the equation, we need to determine the value of the function at $x = 0$, which is the $y$-intercept of the function.

Real-World Applications

The concept of the $y$-intercept has many real-world applications, including:

  • Physics: The $y$-intercept of a function can be used to determine the initial velocity of an object.
  • Engineering: The $y$-intercept of a function can be used to determine the initial current of a circuit.
  • Economics: The $y$-intercept of a function can be used to determine the initial price of a product.

Conclusion

In conclusion, the $y$-intercept of a function is an important concept in mathematics, particularly in algebra and calculus. It is the point at which the graph of the function intersects the $y$-axis, and it can be used to determine the value of the function at a specific point. We have discussed how to find the $y$-intercept of a linear function, and we have used the function $f(x) = 4 - 5x$ as an example. We have shown that the $y$-intercept of this function is $4$.

Q: What is the $y$-intercept of the function $f(x) = 4 - 5x$?

A: The $y$-intercept of the function $f(x) = 4 - 5x$ is $4$.

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

A: The $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. It is the value of $y$ when $x$ is equal to zero.

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

A: To find the $y$-intercept of a function, you can substitute $x = 0$ into the equation of the function and simplify the equation to find the value of the function at $x = 0$.

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

A: The $y$-intercept is the point at which the graph of the function intersects the $y$-axis, while the $x$-intercept is the point at which the graph of the function intersects the $x$-axis.

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

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

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

A: Yes, the $y$-intercept can be a fraction. For example, if the function is $f(x) = \frac{1}{2} - \frac{1}{3}x$, then the $y$-intercept is $\frac{1}{2}$.

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

A: Yes, the $y$-intercept can be a decimal. For example, if the function is $f(x) = 3.5 - 2.1x$, then the $y$-intercept is $3.5$.

Q: What is the $y$-intercept of the function $f(x) = 2x^2 + 3x - 4$?

A: To find the $y$-intercept of the function $f(x) = 2x^2 + 3x - 4$, we can substitute $x = 0$ into the equation of the function and simplify the equation to find the value of the function at $x = 0$.

f(x)=2x2+3x−4f(x) = 2x^2 + 3x - 4

f(0)=2(0)2+3(0)−4f(0) = 2(0)^2 + 3(0) - 4

f(0)=0+0−4f(0) = 0 + 0 - 4

f(0)=−4f(0) = -4

Therefore, the $y$-intercept of the function $f(x) = 2x^2 + 3x - 4$ is $-4$.

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 can substitute $x = 0$ into the equation of the function. However, we need to be careful because the function is not defined at $x = 0$.

f(x)=1xf(x) = \frac{1}{x}

f(0)=10f(0) = \frac{1}{0}

This is undefined, so we cannot find the $y$-intercept of the function $f(x) = \frac{1}{x}$.

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

A: To find the $y$-intercept of the function $f(x) = e^x$, we can substitute $x = 0$ into the equation of the function and simplify the equation to find the value of the function at $x = 0$.

f(x)=exf(x) = e^x

f(0)=e0f(0) = e^0

f(0)=1f(0) = 1

Therefore, the $y$-intercept of the function $f(x) = e^x$ is $1$.

Conclusion

In conclusion, the $y$-intercept of a function is an important concept in mathematics, particularly in algebra and calculus. It is the point at which the graph of the function intersects the $y$-axis, and it can be used to determine the value of the function at a specific point. We have discussed how to find the $y$-intercept of a linear function, and we have used various examples to illustrate the concept.