Introduction to Functions f and g

Oct 13, 2025 by recoveryshake.com 119 views

Introduction to Functions f and g

In mathematics, functions are essential concepts that help us understand the relationship between variables. Two functions, f(x) and g(x), are given, and we need to determine the correct answer based on their characteristics. Function f(x) is represented by the equation f(x) = x^2 - 5x + 6, while function g(x) has a vertex at (2,4) and opens downwards.

Understanding Function f(x)

Function f(x) is a quadratic function, which means it can be represented in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. In this case, the equation f(x) = x^2 - 5x + 6 can be rewritten as f(x) = x^2 - 5x + 6 = (x - 2)(x - 3). This shows that function f(x) has two roots at x = 2 and x = 3.

Understanding Function g(x)

Function g(x) has a vertex at (2,4) and opens downwards. This means that the parabola represented by function g(x) is a downward-facing parabola, which can be represented in the form f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this case, the vertex is (2,4), so the equation of function g(x) can be written as f(x) = a(x - 2)^2 + 4.

Determining the Correct Answer

To determine the correct answer, we need to compare the characteristics of function f(x) and function g(x). Function f(x) has two roots at x = 2 and x = 3, while function g(x) has a vertex at (2,4) and opens downwards. Since function g(x) has a vertex at (2,4), it means that the parabola represented by function g(x) passes through the point (2,4). Therefore, the correct answer is that function g(x) passes through the point (2,4).

Key Characteristics of Function f(x)

Function f(x) is a quadratic function.
The equation of function f(x) is f(x) = x^2 - 5x + 6.
Function f(x) has two roots at x = 2 and x = 3.
The parabola represented by function f(x) opens upwards.

Key Characteristics of Function g(x)

Function g(x) has a vertex at (2,4).
The parabola represented by function g(x) opens downwards.
Function g(x) passes through the point (2,4).

Conclusion

In conclusion, function f(x) is a quadratic function with two roots at x = 2 and x = 3, while function g(x) has a vertex at (2,4) and opens downwards. The correct answer is that function g(x) passes through the point (2,4).

Final Answer

The final answer is that function g(x) passes through the point (2,4).

References

[1] "Quadratic Functions." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/quadratic-functions.
[2] "Vertex Form of a Quadratic Function." Math Open Reference, www.mathopenref.com/quadraticvertexform.html.

Additional Resources

[1] "Quadratic Functions." Wolfram MathWorld, Wolfram Research, mathworld.wolfram.com/QuadraticFunction.html.
[2] "Vertex Form of a Quadratic Function." Purplemath, Purplemath, www.purplemath.com/modules/quadvert.htm.

Introduction to Functions f and g

In our previous article, we discussed the characteristics of two functions, f(x) and g(x). Function f(x) is represented by the equation f(x) = x^2 - 5x + 6, while function g(x) has a vertex at (2,4) and opens downwards. In this article, we will answer some frequently asked questions about functions f and g.

Q: What is the equation of function f(x)?

A: The equation of function f(x) is f(x) = x^2 - 5x + 6.

Q: What are the roots of function f(x)?

A: The roots of function f(x) are x = 2 and x = 3.

Q: What is the vertex of function g(x)?

A: The vertex of function g(x) is (2,4).

Q: Does function g(x) open upwards or downwards?

A: Function g(x) opens downwards.

Q: What is the equation of function g(x)?

A: The equation of function g(x) can be written as f(x) = a(x - 2)^2 + 4, where a is a constant.

Q: Does function g(x) pass through the point (2,4)?

A: Yes, function g(x) passes through the point (2,4).

Q: What is the difference between function f(x) and function g(x)?

A: Function f(x) is a quadratic function with two roots at x = 2 and x = 3, while function g(x) has a vertex at (2,4) and opens downwards.

Q: Can function g(x) be represented in the form f(x) = ax^2 + bx + c?

A: No, function g(x) cannot be represented in the form f(x) = ax^2 + bx + c, because it has a vertex at (2,4) and opens downwards.

Q: What is the significance of the vertex of function g(x)?

A: The vertex of function g(x) represents the maximum or minimum point of the parabola represented by function g(x).

Q: Can function f(x) be represented in the vertex form f(x) = a(x - h)^2 + k?

A: No, function f(x) cannot be represented in the vertex form f(x) = a(x - h)^2 + k, because it does not have a vertex.

Conclusion

In conclusion, we have answered some frequently asked questions about functions f and g. Function f(x) is a quadratic function with two roots at x = 2 and x = 3, while function g(x) has a vertex at (2,4) and opens downwards.

Final Answer

The final answer is that function g(x) passes through the point (2,4).

References

[1] "Quadratic Functions." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/quadratic-functions.
[2] "Vertex Form of a Quadratic Function." Math Open Reference, www.mathopenref.com/quadraticvertexform.html.

Additional Resources

[1] "Quadratic Functions." Wolfram MathWorld, Wolfram Research, mathworld.wolfram.com/QuadraticFunction.html.
[2] "Vertex Form of a Quadratic Function." Purplemath, Purplemath, www.purplemath.com/modules/quadvert.htm.