Example Question - function inverse
Here are examples of questions we've helped users solve.
Finding the Inverse of a Function
To find the inverse of the function \( f(x) = \sqrt{x} - 2 \), we need to switch the roles of x and y and then solve for y. Here are the steps: 1. Write the original function with y: \( y = \sqrt{x} - 2 \). 2. Swap x and y: \( x = \sqrt{y} - 2 \). 3. Solve for y: Starting with the equation from step 2, we will isolate y: \[ x = \sqrt{y} - 2 \] \[ x + 2 = \sqrt{y} \] (Add 2 to both sides) Now we need to get rid of the square root by squaring both sides of the equation: \[ (x + 2)^2 = y \] So the inverse function \( f^{-1}(x) \) is: \[ f^{-1}(x) = (x + 2)^2 \]
Finding the Inverse of a Function
To find the inverse of the function \( f(x) = 2(x + 4)^2 - 5 \) for \( x \leq -4 \), we'll follow these steps: 1. Replace \( f(x) \) with \( y \): \[ y = 2(x + 4)^2 - 5 \] 2. Swap \( x \) and \( y \), because the inverse function \( f^{-1}(x) \) will take the output of \( f \) (which we're calling \( y \)) and produce the original input \( x \): \[ x = 2(y + 4)^2 - 5 \] 3. Solve this equation for \( y \) to find \( f^{-1}(x) \): \[ x + 5 = 2(y + 4)^2 \] \[ \frac{x + 5}{2} = (y + 4)^2 \] Now we take the square root of both sides. Since we know that \( x \leq -4 \), that means the function is dealing with the left half of the parabola, where \( y \) must be less than or equal to -4 since it's decreasing at that part. So, we choose the negative square root to maintain the function inverse: \[ \sqrt{\frac{x + 5}{2}} = y + 4 \] \[ y = -4 \pm \sqrt{\frac{x + 5}{2}} \] Since \( y \leq -4 \), we use the negative square root: \[ y = -4 - \sqrt{\frac{x + 5}{2}} \] So the inverse function \( f^{-1}(x) \) is: \[ f^{-1}(x) = -4 - \sqrt{\frac{x + 5}{2}} \] To check the solution algebraically, you would compose \( f \) and \( f^{-1} \) and ensure that the composition equals \( x \) and vice versa. Graphically, the function \( f \) and its inverse \( f^{-1} \) should be reflections of each other across the line \( y = x \). Unfortunately, as an AI, I can't confirm the graphic representation directly, but you can check this by plotting both functions on graph paper or a graphing utility.
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