Answer The Following True Or False:If $f(x$\] And $g(x$\] Are Differentiable Functions Such That The Graphs Of $y=f(x$\] And $y=g(x$\] Each Have An $x$-intercept At $x=9$, Then The Graph Of

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Introduction

In this article, we will explore the concept of differentiable functions and their properties. We will examine the given statement and determine whether it is true or false. The statement claims that if two differentiable functions, $f(x)$ and $g(x)$, have $x$-intercepts at $x=9$, then the graph of their sum, $y=f(x)+g(x)$, also has an $x$-intercept at $x=9$. We will analyze the properties of differentiable functions and their sums to determine the validity of this statement.

Properties of Differentiable Functions

A differentiable function is a function that has a derivative at every point in its domain. The derivative of a function represents the rate of change of the function with respect to its input. Differentiable functions have several important properties, including:

• Continuity: Differentiable functions are continuous at every point in their domain.
• Differentiability: Differentiable functions have a derivative at every point in their domain.
• Uniqueness of Derivatives: If a function has a derivative at a point, then the derivative is unique.

The Sum of Differentiable Functions

The sum of two differentiable functions, $f(x)$ and $g(x)$, is also a differentiable function. The derivative of the sum is given by the following formula:

$$\frac{d}{dx}(f(x)+g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$$

This formula shows that the derivative of the sum is the sum of the derivatives.

The $x$-Intercept of a Function

The $x$-intercept of a function is the point where the function intersects the $x$-axis. In other words, it is the point where the function has a value of zero. If a function has an $x$-intercept at $x=a$, then the function can be written in the form:

$$f(x) = (x-a)h(x)$$

where $h(x)$ is a differentiable function.

The Graph of $y=f(x)+g(x)$

The graph of $y=f(x)+g(x)$ is the sum of the graphs of $y=f(x)$ and $y=g(x)$. If the graphs of $y=f(x)$ and $y=g(x)$ each have an $x$-intercept at $x=9$, then the graph of $y=f(x)+g(x)$ will also have an $x$-intercept at $x=9$.

Proof

To prove this statement, we can use the following argument:

• Let $f(x)$ and $g(x)$ be differentiable functions that have $x$-intercepts at $x=9$.
• Then, we can write $f(x) = (x-9)h(x)$ and $g(x) = (x-9)k(x)$, where $h(x)$ and $k(x)$ are differentiable functions.
• The sum of $f(x)$ and $g(x)$ is given by:

$$f(x)+g(x) = (x-9)h(x) + (x-9)k(x)$$

• Simplifying this expression, we get:

$$f(x)+g(x) = (x-9)(h(x)+k(x))$$

• Since $h(x)$ and $k(x)$ are differentiable functions, their sum is also a differentiable function.
• Therefore, the graph of $y=f(x)+g(x)$ has an $x$-intercept at $x=9$.

Conclusion

In conclusion, the statement that if $f(x)$ and $g(x)$ are differentiable functions that have $x$-intercepts at $x=9$, then the graph of $y=f(x)+g(x)$ also has an $x$-intercept at $x=9$ is TRUE. This result follows from the properties of differentiable functions and their sums.

Final Answer

The final answer is: TRUE

Introduction

In our previous article, we explored the concept of differentiable functions and their properties. We also examined the statement that if two differentiable functions, $f(x)$ and $g(x)$, have $x$-intercepts at $x=9$, then the graph of their sum, $y=f(x)+g(x)$, also has an $x$-intercept at $x=9$. In this article, we will answer some of the most frequently asked questions about differentiable functions and their sums.

Q: What is a differentiable function?

A: A differentiable function is a function that has a derivative at every point in its domain. The derivative of a function represents the rate of change of the function with respect to its input.

Q: What are some examples of differentiable functions?

A: Some examples of differentiable functions include:

• Polynomial functions: Functions of the form $f(x) = ax^n + bx^{n-1} + \ldots + cx + d$, where $a, b, \ldots, c, d$ are constants and $n$ is a positive integer.
• Rational functions: Functions of the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions.
• Trigonometric functions: Functions of the form $f(x) = \sin x, \cos x, \tan x, \ldots$.

Q: What is the derivative of a function?

A: The derivative of a function $f(x)$ is denoted by $f'(x)$ and represents the rate of change of the function with respect to its input. The derivative is calculated using the following formula:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Q: What is the sum of two differentiable functions?

A: The sum of two differentiable functions $f(x)$ and $g(x)$ is also a differentiable function. The derivative of the sum is given by the following formula:

$$\frac{d}{dx}(f(x)+g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$$

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

A: The $x$-intercept of a function is the point where the function intersects the $x$-axis. In other words, it is the point where the function has a value of zero. If a function has an $x$-intercept at $x=a$, then the function can be written in the form:

$$f(x) = (x-a)h(x)$$

where $h(x)$ is a differentiable function.

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

A: To find the $x$-intercept of a function, you can set the function equal to zero and solve for $x$. For example, if you have a function $f(x) = x^2 - 4$, you can set it equal to zero and solve for $x$:

$$x^2 - 4 = 0$$

$$x^2 = 4$$

$$x = \pm 2$$

Q: What is the graph of $y=f(x)+g(x)$?

A: The graph of $y=f(x)+g(x)$ is the sum of the graphs of $y=f(x)$ and $y=g(x)$. If the graphs of $y=f(x)$ and $y=g(x)$ each have an $x$-intercept at $x=9$, then the graph of $y=f(x)+g(x)$ will also have an $x$-intercept at $x=9$.

Q: How do I find the graph of $y=f(x)+g(x)$?

A: To find the graph of $y=f(x)+g(x)$, you can add the graphs of $y=f(x)$ and $y=g(x)$. For example, if you have two functions $f(x) = x^2 - 4$ and $g(x) = x^2 + 2$, you can add their graphs to get the graph of $y=f(x)+g(x)$:

$$y = (x^2 - 4) + (x^2 + 2)$$

$$y = 2x^2 - 2$$

Conclusion

In conclusion, we have answered some of the most frequently asked questions about differentiable functions and their sums. We hope that this article has been helpful in clarifying the concepts of differentiable functions and their sums. If you have any further questions, please don't hesitate to ask.

Final Answer

The final answer is: Differentiable functions and their sums are an important topic in mathematics, and understanding their properties can help you solve a wide range of problems.