true or false? A function has exactly one output for every input

Question

hi can somebody help me with these simple true or false questions I need to learn them for my final exam

 

a. |𝑥| = −𝑥 if 𝑥 is a negative number.
b. A function has exactly one output for every input.
c. A given graph is the graph of a function iff it passes the horizontal line test.
d. Two non-vertical lines are parallel if the product of their slopes is -1.
e. There is polynomial with degree 5 which has exactly 7 zeros.
f. 𝑓(𝑥) = 𝑥
2 + 1 is not a one-to-one function.
g. log(𝑥𝑦) = log 𝑥 + log 𝑦 for all real numbers 𝑥 and 𝑦.
h. sin 𝑥 = 2/3 has exactly two solutions in the interval [0, 𝜋].
i. log 0 = 1
j. The range of the function cos 𝑥 is (−∞, ∞).

Answer 1

a)

 

 

True.

 

| -x | = -(-x)

 

| -x | = x

 

 

The absolute value of any number is always positive.

 

 

b)

 

True.  A function cannot have more than one output for one input, otherwise it fails the vertical line test.

 

 

c)

 

False.  It must pass the vertical line test.

 

 

d)

 

False.  The product of their slopes must be the square the slopes, since parallel have the same slope.  It would be true for perpendicular lines through, since their slopes would be negative reciprocals.

 

 

e)

 

False.  A polynomial of nth degree has at most n roots.  Degree 5 must have at most 5 roots.

 

 

f)

 

Clarify question?

 

 

g)

 

True.  There is not imaginary numbers in a logarithm.  x and y must be both positive or both negative.

 

 

h)

 

True.  Since sine must be a number less than one and has a solution in the first and second quadrants.  The given domain lies within those quadrants.

 

 

i)

 

False.  Log(10)=1

 

Log(0) undefined, since there is no value raise to any number that results in zero.

 

 

j)

 

False.  cos(x) is a continuous wave-like function with an amplitude of 1.  This means the range of cos(x) is [-1, 1].