Find The Following For The Function $f(x) = 3x^2 + 3x - 4$:(a) $f(0$\](b) $f(5$\](c) $f(-5$\](d) $f(-x$\](e) $-f(x$\](f) $f(x+3$\](g) $f(3x$\](h) $f(x+h$\]

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Introduction

In this article, we will be evaluating the function $f(x) = 3x^2 + 3x - 4$ at various points and for different inputs. This will help us understand how the function behaves and how it can be used to solve real-world problems.

Evaluating the Function at Specific Points

(a) Evaluating $f(0)$

To evaluate $f(0)$, we need to substitute $x = 0$ into the function $f(x) = 3x^2 + 3x - 4$. This gives us:

$f(0) = 3(0)^2 + 3(0) - 4$ $f(0) = 0 + 0 - 4$ $f(0) = -4$

So, the value of the function at $x = 0$ is $-4$.

(b) Evaluating $f(5)$

To evaluate $f(5)$, we need to substitute $x = 5$ into the function $f(x) = 3x^2 + 3x - 4$. This gives us:

$f(5) = 3(5)^2 + 3(5) - 4$ $f(5) = 3(25) + 15 - 4$ $f(5) = 75 + 15 - 4$ $f(5) = 86$

So, the value of the function at $x = 5$ is $86$.

(c) Evaluating $f(-5)$

To evaluate $f(-5)$, we need to substitute $x = -5$ into the function $f(x) = 3x^2 + 3x - 4$. This gives us:

$f(-5) = 3(-5)^2 + 3(-5) - 4$ $f(-5) = 3(25) - 15 - 4$ $f(-5) = 75 - 15 - 4$ $f(-5) = 56$

So, the value of the function at $x = -5$ is $56$.

Evaluating the Function for Different Inputs

(d) Evaluating $f(-x)$

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

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

So, the value of the function at $x = -x$ is $3x^2 - 3x - 4$.

(e) Evaluating $-f(x)$

To evaluate $-f(x)$, we need to multiply the function $f(x) = 3x^2 + 3x - 4$ by $-1$. This gives us:

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

So, the value of the function at $x = -f(x)$ is $-3x^2 - 3x + 4$.

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

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

$f(x+3) = 3(x+3)^2 + 3(x+3) - 4$ $f(x+3) = 3(x^2 + 6x + 9) + 3x + 9 - 4$ $f(x+3) = 3x^2 + 18x + 27 + 3x + 5$ $f(x+3) = 3x^2 + 21x + 32$

So, the value of the function at $x = x + 3$ is $3x^2 + 21x + 32$.

(g) Evaluating $f(3x)$

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

$f(3x) = 3(3x)^2 + 3(3x) - 4$ $f(3x) = 3(9x^2) + 9x - 4$ $f(3x) = 27x^2 + 9x - 4$

So, the value of the function at $x = 3x$ is $27x^2 + 9x - 4$.

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

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

$f(x+h) = 3(x+h)^2 + 3(x+h) - 4$ $f(x+h) = 3(x^2 + 2hx + h^2) + 3x + 3h - 4$ $f(x+h) = 3x^2 + 6hx + 3h^2 + 3x + 3h - 4$

So, the value of the function at $x = x + h$ is $3x^2 + 6hx + 3h^2 + 3x + 3h - 4$.

Conclusion

In this article, we have evaluated the function $f(x) = 3x^2 + 3x - 4$ at various points and for different inputs. We have seen how the function behaves and how it can be used to solve real-world problems. The function can be used to model quadratic relationships and can be used to solve problems in fields such as physics, engineering, and economics.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Further Reading

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Functions" by Khan Academy
• [3] "Calculus" by MIT OpenCourseWare
  

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Introduction

In our previous article, we evaluated the function $f(x) = 3x^2 + 3x - 4$ at various points and for different inputs. In this article, we will answer some frequently asked questions about the function and its behavior.

Q&A

Q: What is the value of the function at $x = 0$?

A: The value of the function at $x = 0$ is $-4$. This can be found by substituting $x = 0$ into the function $f(x) = 3x^2 + 3x - 4$.

Q: How do I evaluate the function at $x = 5$?

A: To evaluate the function at $x = 5$, substitute $x = 5$ into the function $f(x) = 3x^2 + 3x - 4$. This gives us $f(5) = 3(5)^2 + 3(5) - 4 = 86$.

Q: What is the value of the function at $x = -5$?

A: The value of the function at $x = -5$ is $56$. This can be found by substituting $x = -5$ into the function $f(x) = 3x^2 + 3x - 4$.

Q: How do I evaluate the function for $x = -x$?

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

Q: What is the value of the function at $x = -f(x)$?

A: The value of the function at $x = -f(x)$ is $-3x^2 - 3x + 4$. This can be found by multiplying the function $f(x) = 3x^2 + 3x - 4$ by $-1$.

Q: How do I evaluate the function at $x = x + 3$?

A: To evaluate the function at $x = x + 3$, substitute $x = x + 3$ into the function $f(x) = 3x^2 + 3x - 4$. This gives us $f(x+3) = 3(x+3)^2 + 3(x+3) - 4 = 3x^2 + 21x + 32$.

Q: What is the value of the function at $x = 3x$?

A: The value of the function at $x = 3x$ is $27x^2 + 9x - 4$. This can be found by substituting $x = 3x$ into the function $f(x) = 3x^2 + 3x - 4$.

Q: How do I evaluate the function at $x = x + h$?

A: To evaluate the function at $x = x + h$, substitute $x = x + h$ into the function $f(x) = 3x^2 + 3x - 4$. This gives us $f(x+h) = 3(x+h)^2 + 3(x+h) - 4 = 3x^2 + 6hx + 3h^2 + 3x + 3h - 4$.

Conclusion

In this article, we have answered some frequently asked questions about the function $f(x) = 3x^2 + 3x - 4$ and its behavior. We have seen how the function can be evaluated at various points and for different inputs, and how it can be used to solve real-world problems.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Further Reading

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Functions" by Khan Academy
• [3] "Calculus" by MIT OpenCourseWare