which equation is the inverse of y 9x2 4

Guri Bee

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20-10-2024

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Unveiling the Inverse: Solving for y = 9x² + 4

The question of finding the inverse of a function is a common one in algebra. In this case, we're interested in finding the inverse of the function y = 9x² + 4. But what does it mean to find the inverse?

In simple terms, the inverse of a function "undoes" the original function. If we input a value into the original function and then input the output into its inverse, we should get back the original input. This concept is crucial in many mathematical applications, including solving equations and understanding transformations.

Let's break down the process of finding the inverse of y = 9x² + 4:

1. Swap x and y

The first step is to swap the variables x and y in the original equation. This gives us:

x = 9y² + 4

2. Solve for y

Now, we need to solve this new equation for y. This will give us the equation for the inverse function.

• Isolate the y² term: Subtract 4 from both sides: x - 4 = 9y²

• Divide by 9: (x - 4) / 9 = y²

• Take the square root of both sides: √((x - 4) / 9) = y

• Simplify: y = ±√((x - 4) / 9)

3. The Inverse Equation

We have arrived at our inverse function: y = ±√((x - 4) / 9).

Important Note: The inverse function contains a "±" sign. This signifies that the original function is not one-to-one. A one-to-one function has a unique output for every input. In our case, for every output value, there are two possible input values (one positive and one negative). This is why we need the "±" sign in the inverse function to account for both possibilities.

Visualizing the Inverse:

• The original function y = 9x² + 4 is a parabola opening upwards.
• Its inverse, y = ±√((x - 4) / 9), is also a parabola but opening to the right.
• The graphs of a function and its inverse are symmetric about the line y = x. This symmetry is a direct consequence of the swapping of variables in the inverse finding process.

Applications:

Finding the inverse of a function has many applications in various fields, including:

• Solving equations: For instance, if we want to find the value of x for a given y in the original function, we can use the inverse function to solve for x directly.
• Transformations: Inverse functions are crucial for understanding transformations of graphs. They help us to see how the graph of the original function is reflected, stretched, or shifted when finding its inverse.

Remember: The process of finding an inverse function involves swapping variables and solving for the new variable. Keep in mind that not all functions have inverses, especially those that are not one-to-one.

Key Takeaways:

• The inverse of a function "undoes" the original function.
• Finding the inverse involves swapping variables and solving for the new variable.
• The graphs of a function and its inverse are symmetric about the line y = x.
• Inverse functions have numerous applications in various fields.

Additional Information:

While this article focuses on the specific example of y = 9x² + 4, the process of finding the inverse remains similar for various functions. To delve deeper into this concept, you can explore online resources like Khan Academy or your textbook. You can also consult a math tutor or your teacher for further clarification and practice.