Example Question - inverse function
Here are examples of questions we've helped users solve.
Finding the Inverse of a Function
<p>1. Start with the function \( f(x) = \frac{x + 3}{x - 1} \).</p> <p>2. Replace \( f(x) \) with \( y \): \( y = \frac{x + 3}{x - 1} \).</p> <p>3. Swap \( x \) and \( y \): \( x = \frac{y + 3}{y - 1} \).</p> <p>4. Multiply both sides by \( y - 1 \): \( x(y - 1) = y + 3 \).</p> <p>5. Distribute: \( xy - x = y + 3 \).</p> <p>6. Rearrange to solve for \( y \): \( xy - y = x + 3 \).</p> <p>7. Factor out \( y \): \( y(x - 1) = x + 3 \).</p> <p>8. Divide by \( x - 1 \): \( y = \frac{x + 3}{x - 1} \).</p> <p>9. Therefore, the inverse function is \( f^{-1}(x) = \frac{x + 3}{x - 1} \).</p>
Finding the Inverse of an Exponential Function
<p>Let \( y = e^{4x} - 5 \)</p> <p>Swap x and y to find the inverse: \( x = e^{4y} - 5 \)</p> <p>Add 5 to both sides: \( x + 5 = e^{4y} \)</p> <p>Take the natural logarithm of both sides: \( \ln(x + 5) = \ln(e^{4y}) \)</p> <p>Use the property of logarithms: \( \ln(x + 5) = 4y \)</p> <p>Divide by 4: \( y = \frac{1}{4}\ln(x + 5) \)</p> <p>The inverse function is \( f^{-1}(x) = \frac{1}{4}\ln(x + 5) \)</p>
Finding the Inverse of a Quadratic Function
<p>Sea \( g(x) = x^2 - 1 \) la función dada, y queremos encontrar \( g^{-1}(x) \).</p> <p>Para encontrar la función inversa, primero reemplazamos \( g(x) \) por \( y \):</p> <p>\( y = x^2 - 1 \)</p> <p>Luego, resolvemos para \( x \) en términos de \( y \):</p> <p>\( y + 1 = x^2 \)</p> <p>\( x = \sqrt{y + 1} \), pero dado que el dominio está restringido a los números reales positivos, asumimos que \( x \) también es positivo.</p> <p>Ahora intercambiamos \( x \) e \( y \) para obtener la función inversa:</p> <p>\( y = \sqrt{x + 1} \)</p> <p>Por lo tanto, la expresión algebraica de \( g^{-1}(x) \) es \( \sqrt{x + 1} \).</p>
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 of Exponential Function
To find the inverse of the exponential function \( y = 4^x \), you need to solve for x in terms of y. Starting with the original function: \[ y = 4^x \] Swap the roles of x and y to begin finding the inverse function: \[ x = 4^y \] Now, solve for y by taking the logarithm with base 4 of both sides (since 4 is the base of the exponential function): \[ \log_4(x) = \log_4(4^y) \] By the properties of logarithms, \( \log_b(b^a) = a \), so: \[ \log_4(x) = y \] Therefore, the inverse function is: \[ y = \log_4(x) \] Looking at the options provided: A. \( y = x^{-4} \) - Incorrect, because this represents a power function, not a logarithmic function. B. \( y = (\frac{1}{4})^x \) - Incorrect, this is another exponential function, not the inverse of \( 4^x \). C. \( y = \log_4(x) \) - Correct, as proved above. D. \( y = - \log_4(x) \) - Incorrect, this is the negative of the logarithmic function. The correct answer is C: \( y = \log_4(x) \).
Inverse of Exponential Function
The question asks for the inverse of the exponential function \( y = 4^x \). To find the inverse function, we typically swap \( x \) and \( y \) and then solve for \( y \). Starting with \( y = 4^x \), we swap \( x \) and \( y \) to get \( x = 4^y \). Now we need to solve for \( y \). To do this, we take the logarithm base 4 of both sides: \( \log_4(x) = \log_4(4^y) \). Using the property of logarithms that \( \log_b(b^a) = a \), we simplify the right side to get: \( \log_4(x) = y \). So, the inverse function of \( y = 4^x \) is \( y = \log_4(x) \). The correct answer is: C. \( y = \log_4(x) \).
Inverse Function of Numbers
The question in the image asks us to determine the images by the inverse function of the given numbers. This means we need to find the multiplicative inverse of each number, which is just 1 divided by the number itself. Here are the inverses: 1. For -4: the inverse is \( \frac{1}{-4} \) or -0.25. 2. For \( \frac{1}{5} \): the inverse is \( \frac{1}{\frac{1}{5}} = 5 \). 3. For \( \frac{10}{5} \): the inverse is \( \frac{1}{\frac{10}{5}} = \frac{1}{2} \) or 0.5. 4. For \( \frac{5}{8} \): the inverse is \( \frac{1}{\frac{5}{8}} = \frac{8}{5} \) or 1.6. So the images by the inverse function are: -0.25, 5, 0.5, and 1.6.
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 Function of a Given Function
To find \( f^{-1}(x) \), the inverse of the function \( f(x) = \sqrt{x} - 10 \), follow these steps: 1. Replace \( f(x) \) with \( y \): \[ y = \sqrt{x} - 10 \] 2. Swap \( x \) and \( y \) to find the inverse: \[ x = \sqrt{y} - 10 \] 3. Solve for \( y \): \[ x + 10 = \sqrt{y} \] Now square both sides to get rid of the square root: \[ (x + 10)^2 = (\sqrt{y})^2 \] \[ x^2 + 20x + 100 = y \] The inverse function \( f^{-1}(x) \) is then: \[ f^{-1}(x) = x^2 + 20x + 100 \]
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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