If F ( X ) = 3 X + 10 X F(x) = 3^x + 10x F ( X ) = 3 X + 10 X And G ( X ) = 4 X − 2 G(x) = 4x - 2 G ( X ) = 4 X − 2 , Find ( F + G ) ( X (f+g)(x ( F + G ) ( X ].A. 3 X + 6 X + 2 3^x + 6x + 2 3 X + 6 X + 2 B. 3 X − 6 X + 2 3^x - 6x + 2 3 X − 6 X + 2 C. 17 X − 2 17x - 2 17 X − 2 D. 3 X + 14 X − 2 3^x + 14x - 2 3 X + 14 X − 2

If $f(x) = 3^x + 10x$ and $g(x) = 4x - 2$, find $(f+g)(x)$

Understanding the Problem

To find the value of $(f+g)(x)$, we need to understand what the notation means. The notation $(f+g)(x)$ represents the sum of the functions $f(x)$ and $g(x)$, evaluated at the point $x$. In other words, we need to add the two functions together, term by term.

The Functions $f(x)$ and $g(x)$

The function $f(x)$ is defined as $f(x) = 3^x + 10x$. This function has two terms: an exponential term $3^x$ and a linear term $10x$.

The function $g(x)$ is defined as $g(x) = 4x - 2$. This function has two terms: a linear term $4x$ and a constant term $-2$.

Finding $(f+g)(x)$

To find $(f+g)(x)$, we need to add the two functions together, term by term. We can do this by adding the corresponding terms of the two functions.

$(f+g)(x) = f(x) + g(x)$ $= (3^x + 10x) + (4x - 2)$

Simplifying the Expression

Now, we can simplify the expression by combining like terms.

$(f+g)(x) = 3^x + 10x + 4x - 2$ $= 3^x + 14x - 2$

Conclusion

Therefore, the value of $(f+g)(x)$ is $3^x + 14x - 2$.

Answer

The correct answer is D. $3^x + 14x - 2$.

Discussion

This problem requires a basic understanding of function notation and the ability to add functions together. It also requires the ability to simplify expressions by combining like terms.

Example

Let's consider an example to illustrate the concept. Suppose we have two functions $f(x) = 2x + 1$ and $g(x) = 3x - 2$. We can find the sum of these functions by adding them together, term by term.

$(f+g)(x) = f(x) + g(x)$ $= (2x + 1) + (3x - 2)$ $= 5x - 1$

Key Concepts

• Function notation
• Adding functions together
• Simplifying expressions by combining like terms

Related Topics

• Function composition
• Function inverses
• Graphing functions

Practice Problems

• Find the sum of the functions $f(x) = x^2 + 2x$ and $g(x) = 3x - 1$.
• Find the difference of the functions $f(x) = 2x + 1$ and $g(x) = 3x - 2$.
• Find the product of the functions $f(x) = x^2 + 2x$ and $g(x) = 3x - 1$.

Solutions

• The sum of the functions $f(x) = x^2 + 2x$ and $g(x) = 3x - 1$ is $(f+g)(x) = x^2 + 5x - 1$.
• The difference of the functions $f(x) = 2x + 1$ and $g(x) = 3x - 2$ is $(f-g)(x) = -x + 3$.
• The product of the functions $f(x) = x^2 + 2x$ and $g(x) = 3x - 1$ is $(fg)(x) = 3x^3 - x^2 + 6x^2 - 2x$.

Conclusion

In this article, we have discussed how to find the sum of two functions, $(f+g)(x)$. We have also provided examples and practice problems to illustrate the concept. We have also discussed related topics and provided solutions to practice problems.
Q&A: Finding the Sum of Two Functions

Q: What is the sum of two functions?

A: The sum of two functions is the result of adding the two functions together, term by term. It is denoted by the notation $(f+g)(x)$.

Q: How do I find the sum of two functions?

A: To find the sum of two functions, you need to add the corresponding terms of the two functions. You can do this by combining like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the terms $2x$ and $3x$, you can combine them by adding the coefficients: $2x + 3x = 5x$.

Q: What if I have a constant term and a variable term?

A: If you have a constant term and a variable term, you can combine them by adding or subtracting the constant term from the variable term. For example, if you have the terms $2x$ and $3$, you can combine them by subtracting the constant term from the variable term: $2x - 3$.

Q: Can I have a negative coefficient?

A: Yes, you can have a negative coefficient. A negative coefficient is a coefficient that is multiplied by a negative sign. For example, if you have the term $-2x$, the coefficient is $-2$.

Q: How do I simplify an expression?

A: To simplify an expression, you need to combine like terms and eliminate any unnecessary parentheses or brackets.

Q: What if I have a fraction or a decimal in my expression?

A: If you have a fraction or a decimal in your expression, you can simplify it by finding a common denominator or by converting the fraction or decimal to a decimal or fraction.

Q: Can I have a function with multiple terms?

A: Yes, you can have a function with multiple terms. For example, the function $f(x) = 2x + 3x - 4$ has three terms.

Q: How do I find the sum of a function with multiple terms?

A: To find the sum of a function with multiple terms, you need to add the corresponding terms of the two functions. You can do this by combining like terms.

Q: What if I have a function with a variable in the exponent?

A: If you have a function with a variable in the exponent, you can simplify it by using the properties of exponents. For example, if you have the function $f(x) = 2^x + 3^x$, you can simplify it by using the property of exponents that states $a^m \cdot a^n = a^{m+n}$.

Q: Can I have a function with a negative exponent?

A: Yes, you can have a function with a negative exponent. A negative exponent is an exponent that is multiplied by a negative sign. For example, if you have the function $f(x) = 2^{-x}$, the exponent is $-x$.

Q: How do I simplify a function with a negative exponent?

A: To simplify a function with a negative exponent, you need to use the property of exponents that states $a^{-m} = \frac{1}{a^m}$. For example, if you have the function $f(x) = 2^{-x}$, you can simplify it by using the property of exponents: $f(x) = \frac{1}{2^x}$.

Q: Can I have a function with a fraction or a decimal in the exponent?

A: Yes, you can have a function with a fraction or a decimal in the exponent. For example, if you have the function $f(x) = 2^{1/2} + 3^{1/3}$, the exponents are $1/2$ and $1/3$.

Q: How do I simplify a function with a fraction or a decimal in the exponent?

A: To simplify a function with a fraction or a decimal in the exponent, you need to use the properties of exponents. For example, if you have the function $f(x) = 2^{1/2} + 3^{1/3}$, you can simplify it by using the property of exponents that states $a^{m/n} = \sqrt[n]{a^m}$.

Q: Can I have a function with multiple variables?

A: Yes, you can have a function with multiple variables. For example, the function $f(x,y) = 2x + 3y$ has two variables.

Q: How do I find the sum of a function with multiple variables?

A: To find the sum of a function with multiple variables, you need to add the corresponding terms of the two functions. You can do this by combining like terms.

Q: What if I have a function with a variable in the exponent and multiple variables?

A: If you have a function with a variable in the exponent and multiple variables, you can simplify it by using the properties of exponents and combining like terms.

Q: Can I have a function with a negative exponent and multiple variables?

A: Yes, you can have a function with a negative exponent and multiple variables. For example, the function $f(x,y) = 2^{-x} + 3^{-y}$ has a negative exponent and two variables.

Q: How do I simplify a function with a negative exponent and multiple variables?

A: To simplify a function with a negative exponent and multiple variables, you need to use the property of exponents that states $a^{-m} = \frac{1}{a^m}$ and combine like terms.

Q: Can I have a function with a fraction or a decimal in the exponent and multiple variables?

A: Yes, you can have a function with a fraction or a decimal in the exponent and multiple variables. For example, the function $f(x,y) = 2^{1/2} + 3^{1/3}$ has a fraction in the exponent and two variables.

Q: How do I simplify a function with a fraction or a decimal in the exponent and multiple variables?

A: To simplify a function with a fraction or a decimal in the exponent and multiple variables, you need to use the properties of exponents and combine like terms.

Conclusion

In this article, we have discussed how to find the sum of two functions, $(f+g)(x)$. We have also provided examples and practice problems to illustrate the concept. We have also discussed related topics and provided solutions to practice problems.