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Section 1.2 #56: Find the domain of the function f(x)=8x+4.
Solution: The domain of this function can be found by considering where the denominator equals zero. So we require the denominator to be nonzero, i.e. x+4≠0. Isolating x shows us that x≠−4. Therefore the domain of f is R∖{−4} (or written as (−∞,−4)∪(−4,∞) or written as {x:xisarealnumberandx≠−4} or written as "all real numbers except x=−4" or written as "−∞<x<−4 or −4<x<∞").
Section 1.2 #66: Find the domain and range of the function whose graph is pictured:
Solution: The domain is the set of inputs which have an output. That means we are looking for where on the x-axis has a point shaded in above or below it, i.e. the interval [−2π,2π]. You may also write this domain in the following ways: 2π≤x≤2π or "all real numbers between −2π and 2π, including both −2π and 2π".
The range is the set of numbers which are actually outputted by the function. To find it, look up and down the y-axis for the places which have a shaded dot to the left or right of the y-axis -- i.e. [−1,1].
Section 1.3 #6: Find the slope of the line containing the points (−3,1) and (3,−5).
Solution: Recall the slope of the line containing the points (x1,y1) and (x2,y2) is slope=m=y2−y1x2−x1. Therefore we compute
slope=m=−5−13−(−3)=−66=−1.
Section 1.3 #32: Find the slope of the graph of the linear equation y=−25x+7.
Solution: This equation is already in slope-intercept form. Therefore we may "read off" the slope as the coefficient of x, i.e. the slope is −25.
Section 1.3 #55: Find the slope and the y-intercept of the line with equation 3x+2y=10.
Solution: This equation is not in "slope-intercept" form, so it is not as straightforward as #32 to answer the question. One way to proceed is to algebraically manipulate the equation into slope-intercept form: start with
3x+2y=10,
subtract 3x from both sides to get
2y=−3x+10,
divide both sides by 2 to get
y=−32x+5.
This equation is equivalent to the original and is in slope-intercept form. Therefore we see the slope is −32 and the y-intercept is (0,5). (note: if you ask WolframAlpha to find the y-intercept, it only reports "5" -- their definition of y-intercept is not as a point like ours is...be careful!)