91Ó°ÊÓ

Assume that the domain of \(f\) is the set \(A=\\{-2,-1,0,1,2\\}\). Determine the set of ordered pairs that represents the function \(f\). $$ f(x)=x^{2} $$

(a) Given a function \(f,\) prove that \(g(x)\) is even and \(h(x)\) is odd, where \(g(x)=\frac{1}{2}[f(x)+f(-x)]\) and \(h(x)=\frac{1}{2}[f(x)-f(-x)]\) (b) Use the result of part (a) to prove that any function can be written as a sum of even and odd functions. (c) Use the result of part (b) to write each function as a sum of even and odd functions. \(f(x)=x^{2}-2 x+1, \quad k(x)=\frac{1}{x+1}\)

Determine whether the statement is true or false. Justify your answer. The graphs of \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical.

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. \(L_{1}:(0,-1),(5,9)\) \(L_{2}:(0,3),(4,1)\)

When plotting points on the rectangular coordinate system, is it true that the scales on the \(x\) - and \(y\) -axes must be the same? Explain.

A hospital purchases a new magnetic resonance imaging (MRI) machine for $$\$ 500,000$$. The depreciated value \(y\) (reduced value) after \(t\) years is given by \(y=500,000-40,000 t, 0 \leq t \leq 8\). Sketch the graph of the equation.

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ f(x)=x^{6}-2 x^{2}+3 $$

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ h(x)=x^{3}-5 $$

Consider the functions \(f(x)=x^{2}\) and \(g(x)=\sqrt{x}\). (a) Find \(f / g\) and its domain. (b) Find \(f \circ g\) and \(g \circ f .\) Find the domain of each composite function. Are they the same? Explain.

You purchase an all-terrain vehicle (ATV) for $$\$ 8000$$. The depreciated value \(y\) after \(t\) years is given by \(y=8000-900 t, 0 \leq t \leq 6 .\) Sketch the graph of the equation.