91Ó°ÊÓ

Evaluate the function as indicated. Determine its domain and range. \(f(x)=\left\\{\begin{array}{l}|x|+1, x<1 \\ -x+1, x \geq 1\end{array}\right.\) (a) \(f(-3)\) (b) \(f(1)\) (c) \(f(3)\) (d) \(f\left(b^{2}+1\right)\)

Test for symmetry with respect to each axis and to the origin. $$y=\left|x^{3}+x\right|$$

A student who commutes 27 miles to attend college remembers, after driving a few minutes, that a term paper that is due has been forgotten. Driving faster than usual, the student returns home, picks up the paper, and once again starts toward school. Sketch a possible graph of the student's distance from home as a function of time.

Find the composite functions \((f \circ g)\) and \((g \circ f)\). What is the domain of each composite function? Are the two composite functions equal? \(f(x)=\frac{3}{x}\) \(g(x)=x^{2}-1\)

Think About It, find the coordinates of a second point on the graph of a function \(f\) if the given point is on the graph and the function is (a) even and (b) odd. $$\left(-\frac{3}{2}, 4\right)$$

Temperature Conversion Find a linear equation that expresses the relationship between the temperature in degrees Celsius \(C\) and degrees Fahrenheit \(F\). Use the fact that water freezes at \(0^{\circ} \mathrm{C}\) \(\left(32^{\circ} \mathrm{F}\right)\) and boils at \(100^{\circ} \mathrm{C}\left(212^{\circ} \mathrm{F}\right)\). Use the equation to convert \(72^{\circ} \mathrm{F}\) to degrees Celsius.

(a) Prove that if a graph is symmetric with respect to the \(x\) -axis and to the \(y\) -axis, then it is symmetric with respect to the origin. Give an example to show that the converse is not true. (b) Prove that if a graph is symmetric with respect to one axis and to the origin, then it is symmetric with respect to the other axis.

Automobile Aerodynamies The horsepower \(H\) required to overcome wind drag on a certain automobile is approximated by \(H(x)=0.002 x^{2}+0.005 x-0.029, \quad 10 \leq x \leq 100\) where \(x\) is the speed of the car in miles per hour. (a) Use a graphing utility to graph \(H\). (b) Rewrite the power function so that \(x\) represents the speed in kilometers per hour. [Find \(H(x / 1.6) .]\)

Find an equation of the graph that consists of all points \((x, y)\) having the given distance from the origin. (For a review of the Distance Formula, see Appendix D.) The distance from the origin is \(K(K \neq 1)\) times the distance from \((2,0)\).

Prove that the diagonals of a rhombus intersect at right angles. (A rhombus is a quadrilateral with sides of equal lengths.)