Find The Following For The Function F ( X ) = 4 X 2 + 2 X − 2 F(x) = 4x^2 + 2x - 2 F ( X ) = 4 X 2 + 2 X − 2 :(a) F ( 0 F(0 F ( 0 ](b) F ( 2 F(2 F ( 2 ](c) F ( − 2 F(-2 F ( − 2 ](d) F ( − X F(-x F ( − X ](e) − F ( X -f(x − F ( X ](f) F ( X + 3 F(x+3 F ( X + 3 ](g) F ( 4 X F(4x F ( 4 X ](h) F ( X + H F(x+h F ( X + H ](i) $f(0) =

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Introduction

In this article, we will be evaluating the function $f(x) = 4x^2 + 2x - 2$ at various points and for different transformations. This will help us understand how the function behaves under different conditions and how to apply it in real-world scenarios.

Evaluating the Function at Specific Points

(a) Evaluating $f(0)$

To evaluate $f(0)$, we need to substitute $x = 0$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(0) = 4(0)^2 + 2(0) - 2$ $f(0) = 0 + 0 - 2$ $f(0) = -2$

So, the value of the function at $x = 0$ is $-2$.

(b) Evaluating $f(2)$

To evaluate $f(2)$, we need to substitute $x = 2$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(2) = 4(2)^2 + 2(2) - 2$ $f(2) = 4(4) + 4 - 2$ $f(2) = 16 + 4 - 2$ $f(2) = 18$

So, the value of the function at $x = 2$ is $18$.

(c) Evaluating $f(-2)$

To evaluate $f(-2)$, we need to substitute $x = -2$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(-2) = 4(-2)^2 + 2(-2) - 2$ $f(-2) = 4(4) - 4 - 2$ $f(-2) = 16 - 4 - 2$ $f(-2) = 10$

So, the value of the function at $x = -2$ is $10$.

Evaluating the Function for Different Transformations

(d) Evaluating $f(-x)$

To evaluate $f(-x)$, we need to substitute $x = -x$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

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

So, the value of the function at $x = -x$ is $4x^2 - 2x - 2$.

(e) Evaluating $-f(x)$

To evaluate $-f(x)$, we need to multiply the function $f(x) = 4x^2 + 2x - 2$ by $-1$. This gives us:

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

So, the value of the function at $-f(x)$ is $-4x^2 - 2x + 2$.

(f) Evaluating $f(x+3)$

To evaluate $f(x+3)$, we need to substitute $x = x + 3$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(x+3) = 4(x+3)^2 + 2(x+3) - 2$ $f(x+3) = 4(x^2 + 6x + 9) + 2x + 6 - 2$ $f(x+3) = 4x^2 + 24x + 36 + 2x + 4$ $f(x+3) = 4x^2 + 26x + 40$

So, the value of the function at $x+3$ is $4x^2 + 26x + 40$.

(g) Evaluating $f(4x)$

To evaluate $f(4x)$, we need to substitute $x = 4x$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(4x) = 4(4x)^2 + 2(4x) - 2$ $f(4x) = 4(16x^2) + 8x - 2$ $f(4x) = 64x^2 + 8x - 2$

So, the value of the function at $4x$ is $64x^2 + 8x - 2$.

(h) Evaluating $f(x+h)$

To evaluate $f(x+h)$, we need to substitute $x = x + h$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(x+h) = 4(x+h)^2 + 2(x+h) - 2$ $f(x+h) = 4(x^2 + 2hx + h^2) + 2x + 2h - 2$ $f(x+h) = 4x^2 + 8hx + 4h^2 + 2x + 2h - 2$

So, the value of the function at $x+h$ is $4x^2 + 8hx + 4h^2 + 2x + 2h - 2$.

Conclusion

In this article, we have evaluated the function $f(x) = 4x^2 + 2x - 2$ at various points and for different transformations. We have seen how the function behaves under different conditions and how to apply it in real-world scenarios. The results of our evaluations will help us understand the properties of the function and how to use it in mathematical and real-world applications.

Final Answer

The final answer is not a single number, but rather a collection of values and expressions that we have obtained through our evaluations. We have seen how the function $f(x) = 4x^2 + 2x - 2$ behaves at specific points and for different transformations, and we have obtained expressions for the function at these points and transformations.

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Introduction

In our previous article, we evaluated the function $f(x) = 4x^2 + 2x - 2$ at various points and for different transformations. In this article, we will answer some common questions related to the function and its evaluations.

Q1: What is the value of $f(0)$?

A1: The value of $f(0)$ is $-2$. To evaluate $f(0)$, we need to substitute $x = 0$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(0) = 4(0)^2 + 2(0) - 2$ $f(0) = 0 + 0 - 2$ $f(0) = -2$

Q2: What is the value of $f(2)$?

A2: The value of $f(2)$ is $18$. To evaluate $f(2)$, we need to substitute $x = 2$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(2) = 4(2)^2 + 2(2) - 2$ $f(2) = 4(4) + 4 - 2$ $f(2) = 16 + 4 - 2$ $f(2) = 18$

Q3: What is the value of $f(-2)$?

A3: The value of $f(-2)$ is $10$. To evaluate $f(-2)$, we need to substitute $x = -2$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(-2) = 4(-2)^2 + 2(-2) - 2$ $f(-2) = 4(4) - 4 - 2$ $f(-2) = 16 - 4 - 2$ $f(-2) = 10$

Q4: What is the value of $f(-x)$?

A4: The value of $f(-x)$ is $4x^2 - 2x - 2$. To evaluate $f(-x)$, we need to substitute $x = -x$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

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

Q5: What is the value of $-f(x)$?

A5: The value of $-f(x)$ is $-4x^2 - 2x + 2$. To evaluate $-f(x)$, we need to multiply the function $f(x) = 4x^2 + 2x - 2$ by $-1$. This gives us:

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

Q6: What is the value of $f(x+3)$?

A6: The value of $f(x+3)$ is $4x^2 + 26x + 40$. To evaluate $f(x+3)$, we need to substitute $x = x + 3$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(x+3) = 4(x+3)^2 + 2(x+3) - 2$ $f(x+3) = 4(x^2 + 6x + 9) + 2x + 6 - 2$ $f(x+3) = 4x^2 + 24x + 36 + 2x + 4$ $f(x+3) = 4x^2 + 26x + 40$

Q7: What is the value of $f(4x)$?

A7: The value of $f(4x)$ is $64x^2 + 8x - 2$. To evaluate $f(4x)$, we need to substitute $x = 4x$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(4x) = 4(4x)^2 + 2(4x) - 2$ $f(4x) = 4(16x^2) + 8x - 2$ $f(4x) = 64x^2 + 8x - 2$

Q8: What is the value of $f(x+h)$?

A8: The value of $f(x+h)$ is $4x^2 + 8hx + 4h^2 + 2x + 2h - 2$. To evaluate $f(x+h)$, we need to substitute $x = x + h$ into the function $f(x) = 4x^2 + 2x - 2$. This gives us:

$f(x+h) = 4(x+h)^2 + 2(x+h) - 2$ $f(x+h) = 4(x^2 + 2hx + h^2) + 2x + 2h - 2$ $f(x+h) = 4x^2 + 8hx + 4h^2 + 2x + 2h - 2$

Conclusion

In this article, we have answered some common questions related to the function $f(x) = 4x^2 + 2x - 2$ and its evaluations. We have seen how to evaluate the function at specific points and for different transformations, and we have obtained expressions for the function at these points and transformations. The results of our evaluations will help us understand the properties of the function and how to use it in mathematical and real-world applications.

Final Answer

The final answer is not a single number, but rather a collection of values and expressions that we have obtained through our evaluations. We have seen how the function $f(x) = 4x^2 + 2x - 2$ behaves at specific points and for different transformations, and we have obtained expressions for the function at these points and transformations.