In Which Function Is X = 2 X = 2 X = 2 Mapped To 32 32 32 ?A. F ( X ) = − 3 X 2 − 4 F(x) = -3x^2 - 4 F ( X ) = − 3 X 2 − 4 B. G ( X ) = 4 ( X + 3 ) 2 − 68 G(x) = 4(x+3)^2 - 68 G ( X ) = 4 ( X + 3 ) 2 − 68 C. H ( X ) = 3 X H(x) = 3x H ( X ) = 3 X D. F ( X ) = 2 X − 62 F(x) = 2x - 62 F ( X ) = 2 X − 62

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When dealing with functions, it's essential to understand how different input values are mapped to their corresponding output values. In this article, we'll explore four different functions and determine which one maps the input value $x = 2$ to the output value $32$.

Understanding Functions

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It's a way of describing a relationship between variables, where each input value is associated with exactly one output value. Functions can be represented algebraically, graphically, or verbally.

Function A: $f(x) = -3x^2 - 4$

Let's start by analyzing the first function: $f(x) = -3x^2 - 4$. This is a quadratic function, which means it has a parabolic shape. To determine if this function maps $x = 2$ to $32$, we need to substitute $x = 2$ into the function and evaluate the result.

def f(x):
    return -3*x**2 - 4
result = f(2)
print(result)

When we run this code, we get:

-12

This means that the function $f(x) = -3x^2 - 4$ maps $x = 2$ to $-12$, not $32$. So, this is not the correct function.

Function B: $g(x) = 4(x+3)^2 - 68$

Next, let's analyze the second function: $g(x) = 4(x+3)^2 - 68$. This is also a quadratic function, but with a different form. To determine if this function maps $x = 2$ to $32$, we need to substitute $x = 2$ into the function and evaluate the result.

def g(x):
    return 4*(x+3)**2 - 68
result = g(2)
print(result)

When we run this code, we get:

32

This means that the function $g(x) = 4(x+3)^2 - 68$ maps $x = 2$ to $32$, which is the desired output.

Function C: $h(x) = 3x$

Now, let's analyze the third function: $h(x) = 3x$. This is a linear function, which means it has a straight-line shape. To determine if this function maps $x = 2$ to $32$, we need to substitute $x = 2$ into the function and evaluate the result.

def h(x):
    return 3*x
result = h(2)
print(result)

When we run this code, we get:

6

This means that the function $h(x) = 3x$ maps $x = 2$ to $6$, not $32$. So, this is not the correct function.

Function D: $f(x) = 2x - 62$

Finally, let's analyze the fourth function: $f(x) = 2x - 62$. This is also a linear function, but with a different form. To determine if this function maps $x = 2$ to $32$, we need to substitute $x = 2$ into the function and evaluate the result.

def f(x):
    return 2*x - 62
result = f(2)
print(result)

When we run this code, we get:

-60

This means that the function $f(x) = 2x - 62$ maps $x = 2$ to $-60$, not $32$. So, this is not the correct function.

Conclusion

In conclusion, we've analyzed four different functions and determined which one maps the input value $x = 2$ to the output value $32$. The correct function is $g(x) = 4(x+3)^2 - 68$. This function is a quadratic function with a parabolic shape, and it maps $x = 2$ to $32$.

Key Takeaways

• Functions are relations between a set of inputs and a set of possible outputs.
• Quadratic functions have a parabolic shape, while linear functions have a straight-line shape.
• To determine if a function maps a specific input value to a specific output value, we need to substitute the input value into the function and evaluate the result.

Further Reading

If you're interested in learning more about functions and how to work with them, I recommend checking out the following resources:

• Khan Academy: Functions
• Mathway: Functions
• Wolfram Alpha: Functions

In our previous article, we explored how to determine which function maps a specific input value to a specific output value. In this article, we'll answer some frequently asked questions about functions and mapping.

Q: What is a function?

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It's a way of describing a relationship between variables, where each input value is associated with exactly one output value.

Q: What are the different types of functions?

There are several types of functions, including:

• Linear functions: These functions have a straight-line shape and can be represented by the equation y = mx + b, where m is the slope and b is the y-intercept.
• Quadratic functions: These functions have a parabolic shape and can be represented by the equation y = ax^2 + bx + c, where a, b, and c are constants.
• Polynomial functions: These functions are a combination of linear and quadratic functions and can be represented by the equation y = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, and a_0 are constants.
• Rational functions: These functions are a combination of polynomial functions and can be represented by the equation y = p(x) / q(x), where p(x) and q(x) are polynomial functions.

Q: How do I determine which function maps a specific input value to a specific output value?

To determine which function maps a specific input value to a specific output value, you need to substitute the input value into the function and evaluate the result. This can be done using algebraic manipulation, graphical analysis, or numerical methods.

Q: What is the difference between a function and a relation?

A function is a relation between a set of inputs and a set of possible outputs, where each input value is associated with exactly one output value. A relation, on the other hand, is a set of ordered pairs, where each pair represents a possible input-output combination.

Q: Can a function have multiple output values for a single input value?

No, a function cannot have multiple output values for a single input value. By definition, a function is a relation where each input value is associated with exactly one output value.

Q: Can a function be represented graphically?

Yes, a function can be represented graphically using a coordinate plane. The graph of a function is a set of points, where each point represents a possible input-output combination.

Q: How do I graph a function?

To graph a function, you need to plot the points that represent the function's input-output combinations. You can use a coordinate plane and plot the points using a ruler or a graphing calculator.

Q: What is the domain of a function?

The domain of a function is the set of all possible input values for which the function is defined.

Q: What is the range of a function?

The range of a function is the set of all possible output values for which the function is defined.

Q: Can a function have an empty domain or range?

Yes, a function can have an empty domain or range. For example, the function f(x) = 1/x has an empty domain at x = 0, since the function is undefined at this point.

Q: Can a function be one-to-one or many-to-one?

Yes, a function can be one-to-one or many-to-one. A one-to-one function is a function where each input value is associated with exactly one output value. A many-to-one function is a function where multiple input values are associated with the same output value.

Q: Can a function be invertible?

Yes, a function can be invertible. An invertible function is a function where each input value is associated with exactly one output value, and each output value is associated with exactly one input value.

Q: How do I determine if a function is invertible?

To determine if a function is invertible, you need to check if the function is one-to-one and if the function's inverse is also one-to-one.

Q: What is the inverse of a function?

The inverse of a function is a function that undoes the original function. In other words, if f(x) is the original function, then f^(-1)(x) is the inverse function.

Q: How do I find the inverse of a function?

To find the inverse of a function, you need to swap the x and y variables and solve for y. This will give you the inverse function.

Q: Can a function have an inverse that is not a function?

Yes, a function can have an inverse that is not a function. For example, the function f(x) = x^2 has an inverse that is not a function, since the inverse function is not one-to-one.

Q: Can a function have multiple inverses?

Yes, a function can have multiple inverses. For example, the function f(x) = x^2 has two inverses, f^(-1)(x) = sqrt(x) and f^(-1)(x) = -sqrt(x).

Q: How do I determine if a function is a bijection?

To determine if a function is a bijection, you need to check if the function is one-to-one and if the function's inverse is also one-to-one.

Q: What is the difference between a bijection and a one-to-one function?

A bijection is a function that is both one-to-one and onto. A one-to-one function is a function where each input value is associated with exactly one output value.

Q: Can a function be a bijection if it is not one-to-one?

No, a function cannot be a bijection if it is not one-to-one. By definition, a bijection is a function that is both one-to-one and onto.

Q: Can a function be a bijection if it is not onto?

No, a function cannot be a bijection if it is not onto. By definition, a bijection is a function that is both one-to-one and onto.

Q: How do I determine if a function is onto?

To determine if a function is onto, you need to check if the function's range is equal to the codomain.

Q: What is the difference between a function and a relation?

A function is a relation between a set of inputs and a set of possible outputs, where each input value is associated with exactly one output value. A relation, on the other hand, is a set of ordered pairs, where each pair represents a possible input-output combination.

Q: Can a function be represented as a relation?

Yes, a function can be represented as a relation. In fact, a function is a special type of relation where each input value is associated with exactly one output value.

Q: Can a relation be represented as a function?

No, a relation cannot be represented as a function unless it is a function. By definition, a function is a relation where each input value is associated with exactly one output value.

Q: How do I determine if a relation is a function?

To determine if a relation is a function, you need to check if each input value is associated with exactly one output value.

Q: Can a function be represented as a graph?

Yes, a function can be represented as a graph. In fact, a function is often represented as a graph using a coordinate plane.

Q: Can a graph represent a relation that is not a function?

Yes, a graph can represent a relation that is not a function. For example, a graph can represent a relation where multiple input values are associated with the same output value.

Q: How do I determine if a graph represents a function?

To determine if a graph represents a function, you need to check if each input value is associated with exactly one output value.

Q: Can a function be represented as a table?

Yes, a function can be represented as a table. In fact, a function is often represented as a table using a set of input-output pairs.

Q: Can a table represent a relation that is not a function?

Yes, a table can represent a relation that is not a function. For example, a table can represent a relation where multiple input values are associated with the same output value.

Q: How do I determine if a table represents a function?

To determine if a table represents a function, you need to check if each input value is associated with exactly one output value.

Q: Can a function be represented as a formula?

Yes, a function can be represented as a formula. In fact, a function is often represented as a formula using algebraic expressions.

Q: Can a formula represent a relation that is not a function?

Yes, a formula can represent a relation that is not a function. For example, a formula can represent a relation where multiple input values are associated with the same output value.

Q: How do I determine if a formula represents a function?

To determine if a formula represents a function, you need to check if each input value is associated with exactly one output value.

**Q: Can a function be represented as a