Let's start by defining f:R→[−4,∞[ according to f(x)=−(2/5)cos(πx)−1, and g:R→R according to g(x)=−3x/ 2. In this assignment we will study the composite function of f and g, which satisfies h(x)=f(g(x)) for all x in its definition set. a) Give the expression for h(x) b) Calculate h(3), h(4) and h(5) . Your answer should not contain any sine or cosine function and should not be in decimal form. c) Print the definition set and target set for h d) Determine the set of values for h e) Is h an injective function? If yes, provide a proof; if no, give a counterexample. f) Is h a surjective function? If yes, provide a proof; if no, give a counterexample. Your solution must be well written and your conclusions clearly formulated. Explain reasoning using both text and mathematical symbols.
Let's start by defining f:R→[−4,∞[ according to f(x)=−(2/5)cos(πx)−1, and g:R→R according to g(x)=−3x/ 2. In this assignment we will study the composite function of f
and g, which satisfies h(x)=f(g(x))
for all x in its definition set.
a) Give the expression for h(x)
b) Calculate h(3), h(4) and h(5)
. Your answer should not contain any sine or cosine function and should not be in decimal form.
c) Print the definition set and target set for h
d) Determine the set of values for h
e) Is h an injective function? If yes, provide a proof; if no, give a counterexample.
f) Is h a surjective function? If yes, provide a proof; if no, give a counterexample.
Your solution must be well written and your conclusions clearly formulated. Explain reasoning using both text and mathematical symbols. Only calculations are not accepted.
Step by step
Solved in 3 steps
c)Consider how we define the definition set of a composition h(x)=f(g(x)).
To get you thinking in the right direction. Imagine that we let
g:Z→R and retains the same definition for f. Now consider the composition f(g(x)). Depends on the definition set of f(g(x))on the definition set of f(g(x)).
? (NOTE! You do not need to answer this in your submission, this paragraph is only for your assistance.)
Again, your reasoning for target amount is rather unclear. Please see definition of target quantity. To help you, consider what the target set is for the following functions
f:R→R where f(x)=x^2
g:R→R+ where g(x)=x^4+x^2.
If we instead look at a composition, how do we determine the target amount?.
d) The value set is all numbers that h assumes. But what values are they?
e) It is true that the periodic nature of the cosine function makes it non-injective. However, the period is not 2π, why also h(0)=h(2π) is incorrect.
f) That a function is surjective means that its target set and value set coincide. From c
do you know that the target set for his [−4,∞). You have not figured out the amount of value yet. (NOTE! target set and value set are two different things.)
Hint: If h were surjective, it could therefore assume all values on [−4,∞), is it true?
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