Each Exercise gives a formula for a function y = f(x). In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}.$ As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$ $f(x)=1 / x^{2}, \quad x>0$

Consider an elastic string of length L whose ends are held fixed. The string is set in motion with no initial velocity from an initial position u(x,0)=f(x). carry out the following steps. Let L=10 and a=1 in parts (b) through (d).(a) Find the displacement u(x,t)for the given initial position f(x).(b) Plot u(x,t) versus x for 0≤x≤10 and for several values of t between t=0 and t=20.(c) Plot u(x,t) versus t for 0≤t≤20 and for several values of x.(d) Construct an animation of the solution in time for at least one period.(e) Describe the motion of the string in a few sentences. f(x)=2x/L,0≤x≤L/2,2(L−x)/L,L/2<x≤L

Graph the function $f(x)=1 / x$. What symmetry does the graph have?

Write the equation of a line with a slope of $3$ and a $y$-intercept of $1.$

Determine from its graph if the function is one-to-one.

$$f(x)=\left\{\begin{array}{ll} 1-\frac{x}{2}, & x \leq 0 \\ \frac{x}{x+2}, & x>0 \end{array}\right.$$

Graph the function $f(x)=1 / x$. Show that $f$ is its own inverse.

Simplify the expression: $52 – [30 – 2(7 – x)]$.

What is the inverse of the function $f(x) = x + 2?$

Determine from its graph if the function is one-to-one.

$$f(x)=\left\{\begin{array}{ll} 2 x+6, & x \leq-3 \\ x+4, & x>-3 \end{array}\right.$$

Find the inverse of $f(x)=-x+b$ ( $b$ constant $)$. What angle does the line $y=-x+b$ make with the line $y=x$ ? What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line $y=x$ ?

Find the inverse of $f(x)=-x+b$ ( $b$ constant $)$. What angle does the line $y=-x+b$ make with the line $y=x$ ?

Write the equation of the line in slope intercept form when the slope is $3/7$ and the the line passes through the point $(-14,2)$.

The radius of a circular disk is given as $24$ cm with a maximal error in measurement of $0.2$ cm. Use differentials to estimate the following:

(a) The maximum error in the calculated area of the disk.

(b) The relative maximum error.

(c) The percentage error in that case.

Use matrices to solve the system of equations:

$$\begin{cases} 17x+25y-10z&=3\\ 10x+15y-6z&=5\\ -8x-12y+5z&=-1. \end{cases}$$