91Ó°ÊÓ

Bore and Stroke The displacement \(D\) of an engine is given by the function $$ D=\frac{\pi}{4} B^{2} \cdot S \cdot N $$ where \(B\) is the bore (the diameter of a cylinder), \(S\) is the stroke (distance the piston moves), and \(N\) is the number of cylinders. The two- cylinder Harley-Davidson Evo engine has a bore of 3.498 in. and a stroke of 4.250 in. Find its displacement.

Compression Ratio The compression ratio \(C R\) for an engine is a function of the bore \(B\), the stroke \(S\), and the clearance volume \(V,\) where $$ C R=1+\frac{\pi B^{2} \cdot S}{4 V} $$ The clearance volume is the volume of air in the cylinder when the piston is at top dead center. Find a formula that expresses the clearance volume \(V\) as a function of \(C R, B,\) and \(S\).

Each of the following functions is invertible. Find the inverse using composition. $$ f(x)=-x^{3}+4 $$

Find the distance between the points (2,-4) and (-3,-6) .

Find the midpoint of the line segment whose endpoints are (4,-8) and (-6,16).

Solve each problem. Volume of a Cube Express the volume of a cube \(V(x)\) as a function of the length of a side \(x\). Express the length of a side of a cube \(S(x)\) as a function of the volume \(x\).

Sketch the graph of each linear equation. Be sure to find and show the \(x\) - and \(y\) -intercepts. $$ x=3 y-90 $$

Each registered nurse at Blue Hills Memorial Hospital is first given a \(5 \%\) cost-of-living raise and then a \(\$ 3000\) merit raise. Write a function that transforms each old salary \(x\) into a new salary \(N(x)\). Does it make any difference in which order these raises are given? Explain.

Explain why \((f \circ g)^{-1}=g^{-1} \circ f^{-1}\) for any invertible functions \(f\) and \(g .\) Discuss any restrictions on the domains and ranges of \(f\) and \(g\) for this equation to be correct.

Find the domain and range for the function \(f(x)=x^{2}+1\).