Chapter 1: Problem 54
Solve the following equations. $$2^{x}=55$$
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Determine whether the following statements are true and give an explanation or a counterexample. a. All polynomials are rational functions, but not all rational functions are polynomials. b. If \(f\) is a linear polynomial, then \(f \circ f\) is a quadratic polynomial. c. If \(f\) and \(g\) are polynomials, then the degrees of \(f \circ g\) and \(g \circ f\) are equal. d. To graph \(g(x)=f(x+2),\) shift the graph of \(f\) two units to the right.
Simplify the difference quotients \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) by rationalizing the numerator. $$f(x)=\sqrt{x}$$
The floor function, or greatest integer function, \(f(x)=\lfloor x\rfloor,\) gives the greatest integer less than or equal to \(x\) Graph the floor function, for \(-3 \leq x \leq 3\)
Prove that the area of a sector of a circle of radius \(r\) associated with a central angle \(\theta\) (measured in radians) is \(A=\frac{1}{2} r^{2} \theta\)
The surface area of a sphere of radius \(r\) is \(S=4 \pi r^{2} .\) Solve for \(r\) in terms of \(S\) and graph the radius function for \(S \geq 0\)
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