Substituting Points into Each Function

Oct 13, 2025 by windailys.com 147 views

Understanding the Problem

To determine the correct function that passes through the points $$(2,3)$$ and $$(4,4)$$, we need to substitute these points into each of the given functions and check if they satisfy the equation. This process will help us identify the correct function that passes through the given points.

Substituting Points into Each Function

Function A: $$y=\frac{2}{3} x+2$$

To check if this function passes through the points $$(2,3)$$ and $$(4,4)$$, we substitute these points into the equation.

For point $$(2,3)$$:

y = \frac{2}{3} x + 2

3 = \frac{2}{3} (2) + 2

3 = \frac{4}{3} + 2

3 = \frac{4}{3} + \frac{6}{3}

3 = \frac{10}{3}

Since $$3 \neq \frac{10}{3}$$, function A does not pass through the point $$(2,3)$$.

For point $$(4,4)$$:

y = \frac{2}{3} x + 2

4 = \frac{2}{3} (4) + 2

4 = \frac{8}{3} + 2

4 = \frac{8}{3} + \frac{6}{3}

4 = \frac{14}{3}

Since $$4 \neq \frac{14}{3}$$, function A does not pass through the point $$(4,4)$$.

Function B: $$y=x+4$$

To check if this function passes through the points $$(2,3)$$ and $$(4,4)$$, we substitute these points into the equation.

For point $$(2,3)$$:

y = x + 4

3 = 2 + 4

3 = 6

Since $$3 \neq 6$$, function B does not pass through the point $$(2,3)$$.

For point $$(4,4)$$:

y = x + 4

4 = 4 + 4

4 = 8

Since $$4 \neq 8$$, function B does not pass through the point $$(4,4)$$.

Function C: $$y=\frac{1}{2} x+2$$

To check if this function passes through the points $$(2,3)$$ and $$(4,4)$$, we substitute these points into the equation.

For point $$(2,3)$$:

y = \frac{1}{2} x + 2

3 = \frac{1}{2} (2) + 2

3 = 1 + 2

3 = 3

Since $$3 = 3$$, function C passes through the point $$(2,3)$$.

For point $$(4,4)$$:

y = \frac{1}{2} x + 2

4 = \frac{1}{2} (4) + 2

4 = 2 + 2

4 = 4

Since $$4 = 4$$, function C passes through the point $$(4,4)$$.

Function D: $$y=2 x+3$$

To check if this function passes through the points $$(2,3)$$ and $$(4,4)$$, we substitute these points into the equation.

For point $$(2,3)$$:

y = 2 x + 3

3 = 2 (2) + 3

3 = 4 + 3

3 = 7

Since $$3 \neq 7$$, function D does not pass through the point $$(2,3)$$.

For point $$(4,4)$$:

y = 2 x + 3

4 = 2 (4) + 3

4 = 8 + 3

4 = 11

Since $$4 \neq 11$$, function D does not pass through the point $$(4,4)$$.

Function E: $$y=2 x-1$$

To check if this function passes through the points $$(2,3)$$ and $$(4,4)$$, we substitute these points into the equation.

For point $$(2,3)$$:

y = 2 x - 1

3 = 2 (2) - 1

3 = 4 - 1

3 = 3

Since $$3 = 3$$, function E passes through the point $$(2,3)$$.

For point $$(4,4)$$:

y = 2 x - 1

4 = 2 (4) - 1

4 = 8 - 1

4 = 7

Since $$4 \neq 7$$, function E does not pass through the point $$(4,4)$$.

Conclusion

Based on the analysis, we can conclude that the correct function that passes through the points $$(2,3)$$ and $$(4,4)$$ is:

Function C: $$y=\frac{1}{2} x+2$$

This function satisfies the equation for both points, making it the correct answer.

Q: What is the process of determining the correct function that passes through given points?

A: The process involves substituting the given points into each of the functions and checking if they satisfy the equation. This process helps identify the correct function that passes through the given points.

Q: Why is it important to check if a function passes through a point?

A: Checking if a function passes through a point is crucial in determining the correctness of the function. If a function does not pass through a point, it means that the function is not the correct one.

Q: What are some common mistakes to avoid when determining the correct function?

A: Some common mistakes to avoid include:

Not substituting the points into the equation correctly
Not checking if the function satisfies the equation for both points
Not considering the possibility of multiple functions passing through the points

Q: How can I determine if a function is linear or non-linear?

A: A function is linear if it can be written in the form y = mx + b, where m is the slope and b is the y-intercept. If a function cannot be written in this form, it is non-linear.

Q: What is the significance of the slope in a linear function?

A: The slope in a linear function represents the rate of change of the function. It indicates how much the function changes for a given change in the input variable.

Q: Can a function have multiple points that satisfy the equation?

A: Yes, a function can have multiple points that satisfy the equation. However, if a function has multiple points that satisfy the equation, it means that the function is not unique and there may be multiple functions that pass through the points.

Q: How can I determine if a function is a linear function or a non-linear function?

A: To determine if a function is a linear function or a non-linear function, you can use the following methods:

Check if the function can be written in the form y = mx + b
Check if the function has a constant slope
Check if the function has a constant rate of change

Q: What is the difference between a linear function and a non-linear function?

A: A linear function is a function that can be written in the form y = mx + b, where m is the slope and b is the y-intercept. A non-linear function is a function that cannot be written in this form.

Q: Can a non-linear function have a constant slope?

A: No, a non-linear function cannot have a constant slope. A non-linear function typically has a variable slope, which means that the rate of change of the function changes as the input variable changes.

Q: How can I determine if a function is a quadratic function or a non-quadratic function?

A: To determine if a function is a quadratic function or a non-quadratic function, you can use the following methods:

Check if the function can be written in the form y = ax^2 + bx + c
Check if the function has a constant rate of change
Check if the function has a constant slope

Q: What is the significance of the coefficient of x^2 in a quadratic function?

A: The coefficient of x^2 in a quadratic function represents the rate of change of the function. It indicates how much the function changes for a given change in the input variable.

Q: Can a quadratic function have a constant slope?

A: No, a quadratic function cannot have a constant slope. A quadratic function typically has a variable slope, which means that the rate of change of the function changes as the input variable changes.

Q: How can I determine if a function is a polynomial function or a non-polynomial function?

A: To determine if a function is a polynomial function or a non-polynomial function, you can use the following methods:

Check if the function can be written in the form y = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0
Check if the function has a constant rate of change
Check if the function has a constant slope

Q: What is the significance of the degree of a polynomial function?

A: The degree of a polynomial function represents the highest power of the input variable. It indicates the rate of change of the function.

Q: Can a polynomial function have a constant slope?

A: No, a polynomial function cannot have a constant slope. A polynomial function typically has a variable slope, which means that the rate of change of the function changes as the input variable changes.

Q: How can I determine if a function is a rational function or a non-rational function?

A: To determine if a function is a rational function or a non-rational function, you can use the following methods:

Check if the function can be written in the form y = f(x) / g(x)
Check if the function has a constant rate of change
Check if the function has a constant slope

Q: What is the significance of the numerator and denominator in a rational function?

A: The numerator and denominator in a rational function represent the rate of change of the function. The numerator represents the rate of change of the function, while the denominator represents the rate of change of the function.

Q: Can a rational function have a constant slope?

A: No, a rational function cannot have a constant slope. A rational function typically has a variable slope, which means that the rate of change of the function changes as the input variable changes.

Q: How can I determine if a function is a trigonometric function or a non-trigonometric function?

A: To determine if a function is a trigonometric function or a non-trigonometric function, you can use the following methods:

Check if the function can be written in the form y = sin(x) or y = cos(x)
Check if the function has a constant rate of change
Check if the function has a constant slope

Q: What is the significance of the trigonometric functions in mathematics?

A: The trigonometric functions, such as sin(x) and cos(x), are used to describe the relationships between the angles and sides of triangles. They are also used to model periodic phenomena, such as the motion of objects.

Q: Can a trigonometric function have a constant slope?

A: No, a trigonometric function cannot have a constant slope. A trigonometric function typically has a variable slope, which means that the rate of change of the function changes as the input variable changes.

Q: How can I determine if a function is an exponential function or a non-exponential function?

A: To determine if a function is an exponential function or a non-exponential function, you can use the following methods:

Check if the function can be written in the form y = a^x
Check if the function has a constant rate of change
Check if the function has a constant slope

Q: What is the significance of the base in an exponential function?

A: The base in an exponential function represents the rate of change of the function. It indicates how much the function changes for a given change in the input variable.

Q: Can an exponential function have a constant slope?

A: No, an exponential function cannot have a constant slope. An exponential function typically has a variable slope, which means that the rate of change of the function changes as the input variable changes.

Q: How can I determine if a function is a logarithmic function or a non-logarithmic function?

A: To determine if a function is a logarithmic function or a non-logarithmic function, you can use the following methods:

Check if the function can be written in the form y = log(x)
Check if the function has a constant rate of change
Check if the function has a constant slope

Q: What is the significance of the logarithmic function in mathematics?

A: The logarithmic function is used to describe the relationships between the input and output of a function. It is also used to model phenomena that involve growth or decay.

Q: Can a logarithmic function have a constant slope?

A: No, a logarithmic function cannot have a constant slope. A logarithmic function typically has a variable slope, which means that the rate of change of the function changes as the input variable changes.

Q: How can I determine if a function is a power function or a non-power function?

A: To determine if a function is a power function or a non-power function, you can use the following methods:

Check if the function can be written in the form y = x^n
Check if the function has a constant rate of change
Check if the function has a constant slope

Q: What is the significance of the power function in mathematics?

A: The power function is used to describe the relationships between the input and output of a function. It is also used to model phenomena that involve growth or decay.

Q: Can a power function have a constant slope?

A: No, a power function cannot have a constant slope. A power function typically has a variable slope, which means that the rate of change of the function changes as the input variable changes.

Q: How can I determine if a function is a root function or a non-root function?

A: To determine if a function is a root function or a non-root function, you can use the following methods:

Check if the function