Question

In Exercises 37 and 38 , the derivative of the function has the same sign for all \(x\) in its domain, but the function is not one-to-one. Explain. $$ f(x)=\frac{x}{x^{2}-4} $$

Short answer

The derivative of the function \(f(x) = x/(x^{2}-4)\) is constant and positive across its domain, indicating that the function is always increasing. However, when we plot the function, we see that it's not one-to-one because it doesn't pass the horizontal line test. The function's graph intersects horizontal lines at multiple points, proving that it isn't one-to-one.

Step-by-step solution

Compute the derivative

Use the quotient rule for differentiation to find the derivative of the function. The quotient rule states that for two differentiable functions \(u(x)\) and \(v(x)\), the derivative of \(u/v\) is given by \((v*u' - u*v')/v^2\). Here, \(u(x) = x\) and \(v(x) = x^2 - 4\), so compute the derivative accordingly.

Look for sign changes

Evaluate the derivative and simplify. The sign of this derivative will indicate whether the function is increasing or decreasing. The function will stay constant until the derivative changes its sign. If the derivative has the same sign for all \(x\) in its domain, the function is either entirely increasing or decreasing.

Test for one-to-one

Graph the function and perform the horizontal line test. If the horizontal line intersects the graph at more than one point, then the function isn't one-to-one. A function is only one-to-one if each y-value has a unique corresponding x-value.