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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
- a. A function could have the property that f(−x) = f(x), for all x.
- b. cos (a + b) = cos a + cos b, for all a and b in [0, 2π].
- c. If f is a linear function of the form f(x) = mx + b, then f(u + v) = f(u) + f(v), for all u and v.
- d. The function f(x) = 1 − x has the property that f(f(x)) = x.
- e. The set {x: |x + 3| > 4} can be drawn on the number line without lifting your pencil.
- f. log10(xy) = (log10 x)(log10 y)
- g. sin−1 (sin (2π)) = 0
a.
![Check Mark](/static/check-mark.png)
Whether the given statement is true or not and give an explanation or a counter example.
Answer to Problem 1RE
The statement is true.
Explanation of Solution
Given:
A function could have a property that
Consider the function
Then,
Therefore, the statement is true.
b.
![Check Mark](/static/check-mark.png)
Whether the given statement is true or not and give an explanation or a counter example.
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given:
The identity
Calculation:
Take
Then,
And,
That is,
Therefore, the statement is false.
c.
![Check Mark](/static/check-mark.png)
Whether the given statement is true and give an explanation or a counter example.
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given:
If
Calculation:
Take
That is,
The right hand side of equation
And, the left hand side of equation
Thus,
Therefore, the statement is false.
d.
![Check Mark](/static/check-mark.png)
Whether the given statement is true or not and give an explanation or a counter example.
Answer to Problem 1RE
The statement is true.
Explanation of Solution
Given:
The function
Calculation:
Consider the function
Then,
Therefore, the statement is true.
e.
![Check Mark](/static/check-mark.png)
Whether the given statement is true or not and give an explanation or a counter example.
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given:
The set
Calculation:
The given set
Therefore, the set
Therefore, the statement is false.
f.
![Check Mark](/static/check-mark.png)
Whether the given statement is true or not and give an explanation or a counter example.
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given:
The equation,
Calculation:
Take,
Then,
and,
Then,
Therefore, the statement is false.
g.
![Check Mark](/static/check-mark.png)
Whether the given statement is true or not and give an explanation or a counter example.
Answer to Problem 1RE
The statement is true.
Explanation of Solution
Given:
The equation,
Consider
Then, compute the following.
Thus,
Therefore, the statement is true.
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Chapter 1 Solutions
Calculus: Single Variable, Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
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