Which transformation of a function's graph represents the inverse of the function? A Rotation of 90^circ about the origin B Reflection across the origin C Reflection across the line y=x (D) Reflection across the y-axis

To determine which transformation of a function's graph represents the inverse of the function, we need to understand the relationship between a function and its inverse.<br /><br />The inverse of a function is obtained by swapping the x-coordinates and y-coordinates of each point on the original function's graph. This means that if the original function is represented by the equation y = f(x), then the inverse function will be represented by the equation x = f(y).<br /><br />Now let's analyze each option:<br /><br />A) Rotation of $90^{\circ }$ about the origin: This transformation would not preserve the relationship between x and y, so it does not represent the inverse of the function.<br /><br />B) Reflection across the origin: This transformation would also not preserve the relationship between x and y, so it does not represent the inverse of the function.<br /><br />C) Reflection across the line $y=x$: This transformation would swap the x-coordinates and y-coordinates of each point on the original function's graph, which is exactly what we need to obtain the inverse function. Therefore, this option represents the inverse of the function.<br /><br />D) Reflection across the y-axis: This transformation would not preserve the relationship between x and y, so it does not represent the inverse of the function.<br /><br />Therefore, the correct answer is C) Reflection across the line $y=x$.