Your solution for y looks fine, as far as you took it, but when you simplify the radical you should get ##x^4 - 6x^2 + 8x + 1##, which is different from what you show. Also, -1 is not a root of the equation ####x^4 - 6x^2 + 8x + 1 = 0##.Taylor_1989 said:This is the equation, that I got this morning, $$y=\frac{\left(x^2-1\right)\pm\sqrt{\left(1-x^2\right)^2-4x\left(x-2\right)}}{2x}$$. But this is where I was getting stuck because I could not find values for the quartic >0. This is why I thought this method I was doing was wrong. The values I got for x^4-6x+8x+1=0 was x=-1 and x=-0.544.
It's just a change of coordinates. The u and v axes are at 45 degrees to the x and y axes. Actually it is not a pure rotation, there is also an expansion by a factor of √2. To make it a pure rotation you would have to use u=(x+y)/√2 etc. but for the purposes of sketching the graph you can fix that up later.Taylor_1989 said:
Analyzing a function allows us to understand its behavior and properties, such as its domain and range, extrema, and symmetry. This information is crucial for solving problems and making predictions based on the function.
To find the domain, we must identify all the values of the independent variable (usually denoted as x) that the function can take. The range, on the other hand, is all the possible output values (usually denoted as y) of the function. We can determine the domain and range by looking at the graph or using algebraic techniques.
A local maximum (or minimum) occurs at a specific point on the graph where the function has the highest (or lowest) value in a small neighborhood, but it may not be the highest (or lowest) value of the entire function. A global maximum (or minimum), on the other hand, is the highest (or lowest) value of the entire function.
A function is even if it remains unchanged when its input is replaced by its opposite (e.g., f(-x) = f(x)). A function is odd if it changes sign when its input is replaced by its opposite (e.g., f(-x) = -f(x)). If a function does not exhibit either of these properties, it is considered neither even nor odd.
No, a function can only have one inverse. The inverse of a function is a reflection of the original function over the line y = x. If a function has multiple inverses, it means that it does not pass the horizontal line test and is not a one-to-one function.