Given The Function $f(x) = -(x+1)(x-3$\], Simplify Or Expand It To Determine The Expression.

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Introduction

In mathematics, functions are used to describe relationships between variables. A function can be represented in various forms, including algebraic expressions, graphs, and tables. In this article, we will focus on simplifying and expanding the given function $f(x) = -(x+1)(x-3)$ to determine its expression.

Understanding the Function

The given function is a quadratic function, which is a polynomial of degree two. It is represented in factored form, where the expression is written as a product of two binomials. The function is $f(x) = -(x+1)(x-3)$, where the negative sign is outside the parentheses.

Simplifying the Function

To simplify the function, we need to multiply the two binomials inside the parentheses. This can be done using the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. In this case, we have:

$$-(x+1)(x-3) = -(x^2 - 3x + x - 3)$$

Using the distributive property, we can simplify the expression further:

$$-(x^2 - 3x + x - 3) = -(x^2 - 2x - 3)$$

Expanding the Function

To expand the function, we need to multiply the two binomials inside the parentheses. This can be done using the FOIL method, which stands for First, Outer, Inner, Last. The FOIL method is a technique for multiplying two binomials:

$$-(x+1)(x-3) = -(x \cdot x + x \cdot (-3) + 1 \cdot x + 1 \cdot (-3))$$

Using the FOIL method, we can simplify the expression further:

$$-(x \cdot x + x \cdot (-3) + 1 \cdot x + 1 \cdot (-3)) = -(x^2 - 3x + x - 3)$$

Final Expression

After simplifying and expanding the function, we get the final expression:

$$f(x) = -x^2 + 2x + 3$$

This is the simplified and expanded form of the given function $f(x) = -(x+1)(x-3)$.

Conclusion

In this article, we simplified and expanded the given function $f(x) = -(x+1)(x-3)$ to determine its expression. We used the distributive property and the FOIL method to multiply the two binomials inside the parentheses. The final expression is $f(x) = -x^2 + 2x + 3$. This expression can be used to represent the function in a simplified form.

Applications of the Function

The simplified and expanded form of the function $f(x) = -x^2 + 2x + 3$ has several applications in mathematics and other fields. Some of the applications include:

• Graphing: The function can be graphed using the simplified expression. The graph of the function is a parabola that opens downward.
• Optimization: The function can be used to optimize problems that involve quadratic functions.
• Calculus: The function can be used to find the derivative and integral of the function.

Real-World Examples

The simplified and expanded form of the function $f(x) = -x^2 + 2x + 3$ has several real-world examples. Some of the examples include:

• Projectile Motion: The function can be used to model the trajectory of a projectile under the influence of gravity.
• Economics: The function can be used to model the demand and supply of a product in a market.
• Physics: The function can be used to model the motion of an object under the influence of a force.

Conclusion

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Introduction

In our previous article, we simplified and expanded the function $f(x) = -(x+1)(x-3)$ to determine its expression. In this article, we will answer some frequently asked questions (FAQs) related to the function.

Q: What is the simplified form of the function $f(x) = -(x+1)(x-3)$?

A: The simplified form of the function $f(x) = -(x+1)(x-3)$ is $f(x) = -x^2 + 2x + 3$.

Q: How do I multiply two binomials using the distributive property?

A: To multiply two binomials using the distributive property, you need to multiply each term in the first binomial by each term in the second binomial. For example, to multiply $(x+1)$ and $(x-3)$, you would multiply $x$ by $x$, $x$ by $-3$, $1$ by $x$, and $1$ by $-3$.

Q: What is the FOIL method for multiplying two binomials?

A: The FOIL method is a technique for multiplying two binomials. It stands for First, Outer, Inner, Last, and it involves multiplying the first terms in each binomial, then the outer terms, then the inner terms, and finally the last terms.

Q: How do I use the FOIL method to multiply two binomials?

A: To use the FOIL method, you need to multiply the first terms in each binomial, then the outer terms, then the inner terms, and finally the last terms. For example, to multiply $(x+1)$ and $(x-3)$ using the FOIL method, you would multiply $x$ by $x$, $x$ by $-3$, $1$ by $x$, and $1$ by $-3$.

Q: What is the difference between the distributive property and the FOIL method?

A: The distributive property and the FOIL method are both techniques for multiplying two binomials. The distributive property involves multiplying each term in the first binomial by each term in the second binomial, while the FOIL method involves multiplying the first terms in each binomial, then the outer terms, then the inner terms, and finally the last terms.

Q: When should I use the distributive property and when should I use the FOIL method?

A: You should use the distributive property when you need to multiply two binomials that are not in the form of $(a+b)(c+d)$. You should use the FOIL method when you need to multiply two binomials that are in the form of $(a+b)(c+d)$.

Q: Can I use the distributive property and the FOIL method to multiply three or more binomials?

A: Yes, you can use the distributive property and the FOIL method to multiply three or more binomials. However, it may be more difficult to use these techniques for larger numbers of binomials.

Q: How do I graph the function $f(x) = -x^2 + 2x + 3$?

A: To graph the function $f(x) = -x^2 + 2x + 3$, you need to use a graphing tool or a calculator. You can also use a graphing software or a spreadsheet program to graph the function.

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

A: The derivative of the function $f(x) = -x^2 + 2x + 3$ is $f'(x) = -2x + 2$.

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

A: The integral of the function $f(x) = -x^2 + 2x + 3$ is $F(x) = -\frac{x^3}{3} + x^2 + 3x + C$, where $C$ is the constant of integration.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the function $f(x) = -(x+1)(x-3)$. We covered topics such as the simplified form of the function, the distributive property, the FOIL method, graphing, derivatives, and integrals. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the function.