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Chapter 1: Problem 80
Does \((x-3)^{2}+(y-5)^{2}=0\) represent the equation of a circle? If not, describe the graph of this equation.
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I noticed that the difference quotient is always zero if \(f(x)=c,\) where \(c\) is any constant.
Here is the Federal Tax Rate Schedule \(X\) that specifies the tax owed by a
single taxpayer for a recent year. (TABLE CANNOT COPY)
The preceding tax table can be modeled by a piecewise function, where \(x\)
represents the taxable income of a single taxpayer and \(T(x)\) is the tax owed:
$$T(x)=\left\\{\begin{array}{ccc}
0.10 x & \text { if } & 0
Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}-10 x-6 y-30=0$$
114\. If \(f(x)=x^{2}-4\) and \(g(x)=\sqrt{x^{2}-4},\) then \((f \circ g)(x)=-x^{2}\) and \(\left(f^{\circ} g\right)(5)=-25\) 115\. There can never be two functions \(f\) and \(g\), where \(f \neq g\), for which \((f \circ g)(x)=(g \circ f)(x)\) 116\. If \(f(7)=5\) and \(g(4)=7,\) then \((f \circ g)(4)=35\) 117\. If \(f(x)=\sqrt{x}\) and \(g(x)=2 x-1,\) then \((f \circ g)(5)=g(2)\) 118\. Prove that if \(f\) and \(g\) are even functions, then \(f g\) is also an even function. 119\. Define two functions \(f\) and \(g\) so that \(f^{\circ} g=g \circ f\)
A telephone company offers the following plans. Also given are the piecewise functions that model these plans. Use this information to solve. Plan \(A\) \(\cdot \$ 30\) per month buys 120 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$C(t)=\left\\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 120 \\\30+0.30(t-120) & \text { if } t>120 \end{array}\right. $$ Plan \(B\) \(\cdot \ 40\) per month buys 200 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$ C(t)=\left\\{\begin{array}{ll} 40 & \text { if } 0 \leq t \leq 200 \\\ 40+0.30(t-200) & \text { if } t>200 \end{array}\right. $$ Simplify the algebraic expression in the second line of the piecewise function for plan A. Then use point-plotting to graph the function.
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