Which equation is the inverse of y=100-x2?
y=IN100-x
y=10±~X
y=100±vx
y=±vx-100
The Correct Answer and Explanation is:
To find the inverse of the equation ( y = 100 – x^2 ), we will follow a systematic approach involving algebraic manipulation. The process of finding an inverse function essentially entails swapping the roles of ( x ) and ( y ) and then solving for ( y ).
Step 1: Rewrite the Original Equation
The original equation is:
[
y = 100 – x^2
]
Step 2: Swap ( x ) and ( y )
To find the inverse, we switch ( x ) and ( y ):
[
x = 100 – y^2
]
Step 3: Solve for ( y )
Next, we isolate ( y ) in this equation. We start by moving ( y^2 ) to one side and ( x ) to the other side:
[
y^2 = 100 – x
]
Now, we take the square root of both sides. Since the original function ( y = 100 – x^2 ) produces both positive and negative outputs (due to the ( \pm ) in square roots), we need to express this:
[
y = \pm\sqrt{100 – x}
]
Step 4: Identify the Correct Form
Thus, the inverse function can be represented as:
[
y = 100 \pm \sqrt{x}
]
However, we must be careful with this representation. Since the original function opens downward (as indicated by the negative sign before ( x^2 )), we should define the inverse in a specific domain, typically considering only the principal square root for clarity:
[
y = \sqrt{100 – x} \quad \text{(for the positive branch)}
]
or
[
y = -\sqrt{100 – x} \quad \text{(for the negative branch)}
]
Conclusion
Based on the choices provided:
- Correct Answer: None of the given options matches the inverse derived. The inverse function is ( y = 100 \pm \sqrt{x} ) when we swap and solve the original equation correctly.
This demonstrates the importance of careful algebraic manipulation when working with inverse functions, particularly with quadratic equations, as they can produce multiple values depending on the input range and desired output characteristics.