Question

Find the value of $f\left(-2\right),f\left(-1\right),f\left(0\right),f\left(5\right),f\left(x^2\right),f\left(\frac{1}{x}\right)$for the function $f\left(x\right)=\frac{\left|x\right|}{x}$.

Short answer

The value of function are

$f\left(-2\right)=-1,f\left(-1\right)=-1,f\left(0\right)=\text{undefined},f\left(5\right)=1,f\left(x^2\right)=1,f\left(\frac{1}{x}\right)=1$.

Step-by-step solution

Step 1. Evaluate f−2,f−1,f0 for the given function.

From the given function,

$f\left(x\right)=\frac{\left|x\right|}{x}\mathrm{....}\left(1\right)$

Evaluate the value of function at $x=-2,-1,0$gives,

role="math" localid="1644518088722" $$\begin{gathered} f\left(-2\right)=\frac{|-2|}{-2}\mathrm{....}\left(\text{Substitute}x=-2\text{in}\left(1\right)\right) \\ =-1 \\ f\left(-1\right)=\frac{\left|\left(-1\right)\right|}{\left(-1\right)}\mathrm{....}\left(\text{Substitute}x=-1\text{\hspace{0.17em}in}\left(1\right)\right) \\ =-1 \\ f\left(0\right)=\frac{\left|\left(0\right)\right|}{\left(0\right)}\mathrm{....}\left(\text{Substitute}x=0\text{in}\left(1\right)\right) \\ \left(\text{Whichisundefined}\right) \end{gathered}$$

Step 2. Evaluate f5,fx2,f1x for the given function.

Similarly, evaluate the value of function (1) at$x=5,x^2,\frac{1}{x}$ gives,

role="math" localid="1644518425397" $$\begin{gathered} f\left(5\right)=\frac{\left|\left(5\right)\right|}{\left(5\right)}\mathrm{....}\left(\text{Substitute}x=5\text{in}\left(1\right)\right) \\ =1 \\ f\left(x^2\right)=\frac{\left|\left(x^2\right)\right|}{\left(x^2\right)}\mathrm{....}\left(\text{Substitute}x=x^2\text{\hspace{0.17em}in}\left(1\right)\right) \\ =1 \\ f\left(\frac{1}{x}\right)=\frac{\left|\left(\frac{1}{x}\right)\right|}{\left(\frac{1}{x}\right)}\mathrm{....}\left(\text{Substitute}x=\frac{1}{x}\text{in}\left(1\right)\right) \\ \text{=1} \end{gathered}$$

Step 3. Description of steps.

From the obtained values,

$f\left(-2\right)=-1,f\left(-1\right)=-1,f\left(0\right)=\text{undefined},f\left(5\right)=1,f\left(x^2\right)=1,f\left(\frac{1}{x}\right)=1$.