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1. Find the distance from (1,2) to (−1,3).
Solution: Using the distance formula, we get
d((1,2),(−1,3))=√(1−(−1))2+(2−3)2=√22+(−1)2=√4+1=√5.
2. Find the midpoint of the segment between (−1,1) and (5,3).
Solution: The midpoint is
(−1+52,1+32)=(42,42)=(2,2).
3. Find all x and y intercepts of the equation 3y=6−4x.
Solution: To find x-intercept, set y=0 and solve the resulting equation: 0=6−4x. This yields solution x=32 and hence we have x-intercept (32,0) (remember, intercepts are points!) To find the y-intercept, set x=0 and solve the resulting equation: 3y=6 to get y=2. Hence the y-intercept is (0,2).
4. Let f(x)=2x+3 and g(x)=x2+1. Find f(g(x)) and g(f(x)).
Solution: First compute f(g(x)):
f(g(x))=f(x2+1)=2(x2+1)+3=2x2+2+3=2x2+5.
Now compute g(f(x)):
g(2x+3)=(2x+3)2+1=4x2+12x+9+1=4x2+12x+10.
5. Pictured below is an interval:
Express this interval using
a.) interval notation, and
b.) an inequality.
Solution: In interval notation, we write [1,3). As an inequality we write 1≤x<3.