Example Question - find inverse function
Here are examples of questions we've helped users solve.
Finding the Inverse Function of a Linear Function
Para hallar \( g^{-1}(x) \), queremos encontrar la función inversa de \( g(x) \). La función dada es \( g(x) = -2x + 4 \). Para hallar su inversa, seguimos estos pasos: 1. Cambiamos \( g(x) \) por \( y \): \( y = -2x + 4 \). 2. Intercambiamos \( y \) y \( x \) para comenzar a resolver para \( y \): \( x = -2y + 4 \). 3. Resolvemos para \( y \). Primero, movemos el término constante al otro lado del signo igual para aislar los términos con \( y \): \( x - 4 = -2y \). 4. Dividimos ambos lados por -2 para resolver \( y \): \( y = \frac{x - 4}{-2} \). 5. Simplificamos la expresión: \( y = \frac{-x + 4}{2} \) o \( y = -\frac{1}{2}x + 2 \). Por lo tanto, la función inversa de \( g(x) \), que es \( g^{-1}(x) \), es \( g^{-1}(x) = -\frac{1}{2}x + 2 \).
Finding Inverse Function by Switching Variables
The question asks for the inverse of the function f(x) = (2x+3)/(x+2), evaluated at x = -5. To find the inverse function, f^(-1)(x), you need to switch the roles of x and y in the equation and then solve for y. Here's how: 1. Replace f(x) with y: y = (2x + 3)/(x + 2) 2. Switch x and y: x = (2y + 3)/(y + 2) 3. Solve for y: x(y + 2) = 2y + 3 xy + 2x = 2y + 3 xy - 2y = 3 - 2x y(x - 2) = 3 - 2x y = (3 - 2x) / (x - 2) Now that we have the inverse function, we can plug in x = -5 to find the value of f^(-1)(-5): f^(-1)(-5) = (3 - 2(-5)) / ((-5) - 2) f^(-1)(-5) = (3 + 10) / (-5 - 2) f^(-1)(-5) = 13 / (-7) f^(-1)(-5) = -13/7 So, the value of the inverse function f^(-1) at x = -5 is -13/7.
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: \[ f^{-1}(x) = (x + 2)^2 \]
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