Example Question - finding inverse function
Here are examples of questions we've helped users solve.
Finding the Inverse of a Linear Function
Para resolver la imagen, debemos encontrar la inversa de la función \( g(x) = 2x + 4 \), lo cual se denota como \( g^{-1}(x) \). La inversa de una función básicamente intercambia los roles de las variables dependiente e independiente. Es decir, para la función original \( y = g(x) = 2x + 4 \), si queremos encontrar su inversa, despejamos \( x \) en términos de \( y \): \( y = 2x + 4 \) \( y - 4 = 2x \) \( \frac{y - 4}{2} = x \) La inversa entonces será \( g^{-1}(y) = \frac{y - 4}{2} \), pero generalmente usamos \( x \) en lugar de \( y \) en la notación de la función. Así que la función inversa es: \( g^{-1}(x) = \frac{x - 4}{2} \) Ahora, para encontrar el valor de \( g^{-1}(8) \), simplemente sustituimos \( 8 \) en \( x \) en la función inversa que encontramos: \( g^{-1}(8) = \frac{8 - 4}{2} = \frac{4}{2} = 2 \) Entonces, \( g^{-1}(8) = 2 \).
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.
Finding the Inverse of a Quadratic Function
To find the inverse of the function f(x) = 3x^2 + 5, we'll need to follow these steps: 1. Replace f(x) with y: y = 3x^2 + 5 2. Switch x and y: x = 3y^2 + 5 3. Solve for y in terms of x. Starting with the equation x = 3y^2 + 5, we'll solve for y: x = 3y^2 + 5 x - 5 = 3y^2 (y^2) = (x - 5) / 3 y = ±sqrt((x - 5) / 3) However, since we generally consider the principal branch (or the positive square root) for the inverse function, we will choose the positive square root: y = sqrt((x - 5) / 3) Therefore, the inverse function, f^(-1)(x), is: f^(-1)(x) = sqrt((x - 5) / 3) Do note that the domain of the original function must be restricted to non-negative values of x to ensure that the inverse is a function because the original function f(x) = 3x^2 + 5 is not one-to-one over all real numbers due to the parabola nature of y = x^2. This restriction allows the inverse to pass the horizontal line test.
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