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Stewart’s Calculus 8th Edition, Section 1.1, Question 3

Posted by Dr. Fred Zhang | Jan 16, 2019 8:00:00 PM

Teaching Explanations

 

feature_stewartcalc

This posts contains a Teaching Explanation.

You can buy Calculus by Stewart here.

 

Why You Should Trust Me: I’m Dr. Fred Zhang, and I have a bachelor’s degree in math from Harvard.  I’ve racked up hundreds and hundreds of hours of experience working with students from 5th grade through graduate school, and I’m passionate about teaching.  I’ve read the whole chapter of the text beforehand and spent a good amount of time thinking about what the best explanation is and what sort of solutions I would have wanted to see in the problem sets I assigned myself when I taught. 

 

Question: The graph of a function f is given.

Page in 8th Edition: 19

 

Short Answers:

  1. f(1) = 3
  2. f(-1) ~ -.3
  3. f(x)=1 for x = 0 or 3
  4. f(x)=0 for approximately x=-0.6
  5. The domain of x are real numbers between -2 and 4 (or [-2,4], and the range are real numbers between -1 and 3, or [-1,3].
  6. f is increasing on the interval [-2,1)

 

Homework Answer: Same as Short Answers.

 

Motivated Answers:

The question is giving you the graph of the function f.  This means that to figure out what f(x) is, we need to look at the y-value of the graph at x. 

  1. To figure out f(1), we can take put a ruler vertically (up down) on the graph when x=1 and see how high the graph is, which is the same thing as the y-value of the graph.  We can count boxes on the graph paper to see the y-value is 3.

  2. Just like a), we put a ruler vertically at x=-1, and the graph seems to show a y-value of about -.3 (it could be -0.2 or -0.5, but that’s our best guess by eyeballing it).  This means f(-1)~-0.3

  3. The question wants us to find all values of x where f(x)=1.  Since 1 is the output of f, and the output means to y-values, we can take a ruler, put it horizontally at 1, and look at where the ruler hits the graph.  We see the rule hits the graph two times, once when x is 0, and another time when x = 3.

  4. We do the same thing as c), but put the ruler horizontally at 0, which happens to be the x-axis.  The graph hits the ruler at x=-.6 approximately.

  5. You have to find the domain and range of f.  The domain of any function is all valid inputs, or stated the same way, all valid x-values.  We can see from the graph that the graph spans the x-range of -2 though 4 (we can count boxes).  To write this in interval notation, we write the range is [-2,4].  We use solid brackets here because the graph seems to include the endpoints. The range of f is all valid outputs of f.  Stated the same way, these are all valid y-values of the graph.  We can see the graph spans the y-range of -1 through 3, or [-1,3].

  6. If you look at the graph you can see that f seems to be increasing throughout the first part of it, from x-values of -2 to 1.  Writing this in interval notation, we get [-2,1).  We use a parenthesis ) instead of bracket ] because at the point 1, the function is no longer increasing.

 

Video Solution:

 

 

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Dr. Fred Zhang
About the Author

Fred is co-founder of PrepScholar. He scored a perfect score on the SAT and is passionate about sharing information with aspiring students. Fred graduated from Harvard University with a Bachelor's in Mathematics and a PhD in Economics.



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