Let f(x) =(x+3)/(2x-5) Find the inverse function of f. What is the domain of f? What is the...

Question:

Let {eq}f\left( x \right)=\frac{\left( x+3 \right)}{\left( 2x-5 \right)} {/eq}.

(a). Find the inverse function of {eq}f {/eq}.

(b). What is the domain of {eq}f {/eq}?

(c). What is the domain of the inverse function?

Function:

{eq}f\left(x\right)=\frac{x+3}{2x-5} {/eq}

a. Inverse Function:

{eq}\begin{align*} f\left(x\right)&=\frac{x+3}{2x-5} \\ y&=\frac{x+3}{2x-5} \\ x&=\frac{y+3}{2y-5} \\ x\left(2y-5\right)&=y+3 \\ 2xy-5x&=y+3 \\ 2xy-y&=5x+3 \\ y\left(2x-1\right)&=5x+3 \\ y&=\frac{5x+3}{2x-1} \\ f^{-1}\left(x\right)&=\frac{5x+3}{2x-1} \end{align*} {/eq}

b. What is the domain of the function:

The domain of a rational function depends on the denominator. Then:

{eq}\begin{align*} 2x-5&=0 \\ 2x&=5 \\ x&=\frac{5}{2} \end{align*} {/eq}

Domain: {eq}\left(-\infty \:,\:\frac{5}{2}\right)\cup \left(\frac{5}{2},\:\infty \:\right) {/eq}

c. What is the domain of the inverse function.

The inverse is a rational function too, it means the domain of this function depends on its denominator too.

{eq}\begin{align*} 2x-1&=0 \\ 2x&=1 \\ x&=\frac{1}{2} \end{align*} {/eq}

Domain: {eq}\left(-\infty \:,\:\frac{1}{2}\right)\cup \left(\frac{1}{2},\:\infty \:\right) {/eq}