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Chapter 1: Problem 72

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=\frac{5 x-3}{2 x+5}$$

Short Answer

Expert verified
The function \(f(x) = \frac{5x-3}{2x+5}\) does have an inverse, and it is \(f^{-1}(x) = \frac{3+5x}{5-2x}\) when \(x鈮燶frac{5}{2}\).

Step by step solution

01

Check Existence of the Inverse

To determine whether the function has an inverse, it must first be shown that the function is one-to-one. This would require differentiating the function \(f(x)\), however, it's complicated for this function. If the derivative is always increasing or always decreasing, then the function is one-to-one. Alternatively, in case of such complex functions, a visual inspection of the graph can be made. As we look at the graph of this function, it鈥檚 clear that the function is indeed one-to-one. This means the inverse exits.
02

Find the Inverse Function

To find the inverse function, start by setting \(f(x) = y\), so the equation becomes \(y=\frac{5x-3}{2x+5}\). Then, follow these steps: Swap x and y to get \(x=\frac{5y-3}{2y+5}\). Now the equation needs to be solved for y. Cross-multiply to get rid of the denominator, this yields \(x(2y +5)=5y - 3\), which simplifies to \(2xy + 5x = 5y - 3\). Now isolate the y-terms on one side and the x-term on the other to get \(5y - 2xy = 3 + 5x\). We can see that \(y\) appears in two terms so we can factor \(y\) out to get \(y(5 - 2x) = 3 + 5x\). Finally, solve for \(y\) to get the inverse function: \(y=\frac{3+5x}{5-2x}\). Thus, this is the inverse function assuming \(x鈮燶frac{5}{2}\), because we cannot divide by 0.

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