12.11 Function
Exercise 1
Study the parity of the following functions:
\(f_1(x) = x^2 + 2, f_2(x) = x^3 + 4, f_3(x) = e^x − e^{−x}, f_4(x) = \frac{e^{2x} − 1}{e^{2x} + 1}, f_5(x) = \frac{e^x}{(e^x + 1)^2}\)
Exercise 2
Let \(f, g : \mathbb{R} → \mathbb{R}\) be odd functions. What about the parity of \(f + g, f × g\) and \(f \circ g\)?
odd, even, \(f(g(-x))=f(-g(x))=-f(g(x))\), then odd
Exercise 3
Let \(f : \mathbb{R} → \mathbb{R}\) be an even function. Assume that the restriction of \(f\) to \(\mathbb{R}^-\) is increasing. What can be said about the monotonicity of the restriction of \(f\) to \(\mathbb{R}^+\).
decreasing
Exercise 4
(a) Let \(f : \mathbb{R} → \mathbb{R}, x ↦ x^2\), and let \(A = [−1, 4]\). Determine
(i) the image of \(A\) by \(f\);
\([0,16]\)
(ii) the pre-image of \(A\) by \(f\).
\([-2,2]\)
(b) Consider the function \(\sin : \mathbb{R} → \mathbb{R}\). What is the image, by \(\sin\), of \(\mathbb{R}\)? Of \([0, 2\pi]\)? Of \([0, \pi/2]\)? What is the pre-image, by \(\sin\), of \([0, 1]\)? Of \([3, 4]\)? Of \([1, 2]\)?
\([-1,1],\left\lbrack-1,1\right\rbrack,\left\lbrack0,1\right\rbrack\)
\([0,\pi/2]\ldots,\emptyset,\frac{\pi}{2}\ldots\)
Exercise 5
Let \(x ∈ \mathbb{R}^+\), be \(f(x) = \frac{x}{x+1}\). Determine \(f\circ f\circ\cdot\cdot\cdot\circ f(x)\) (where the symbol \(f\) appears \(n\) times) as a function of \(n ∈ \mathbb{N}^*\) and \(x ∈ \mathbb{R}^+\).
\(f(f(x))=f(\frac{x}{x+1})=\frac{\frac{x}{x+1}}{\frac{x}{x+1}+1}=\frac{x}{2x+1}\)
\(f(f(f(x)))= \frac{\frac{x}{2x+1} }{2\frac{x}{2x+1} +1} =\frac{x}{2x+2x+1}=\frac{x}{4x+1}\)
Thus \(f\circ f\circ...\circ f(x)=\frac{x}{2\left(n-1\right)x+1}\)
Exercise 6
Let \(g:[0,+\infty)\to[0,1)\) be defined by \(g(x) = \frac{x}{1+x}\). Show that \(g\) is bijective and determine its inverse.
\(y=\frac{x}{1+x}\Rightarrow xy-x=-y\Rightarrow x=\frac{y}{1-y}\)
Exercise 7
Show that the function \(f : \mathbb{R} → \mathbb{R}^*_+\) defined by \(f(x) = \frac{e^x + 2}{e^{−x}}\) is bijective. Compute its inverse \(f^{−1}\).
\(y=\frac{e^{x}+2}{e^{-x}}\Rightarrow y-e^{2x}=2e^{x}\Rightarrow t^2+2t-y=0\Rightarrow t=\frac{-2\pm\sqrt{4+4y}}{2}=-1\pm\sqrt{1+y}\Rightarrow x=\ln\left(\sqrt{1+y}-1\right)\)
Exercise 8
Let \(E,F\) be two sets and \(f : E → F\). Let \(A ⊂ E\) and \(B ⊂ F\). Prove the equivalence:
\(f(A)\cap B=\emptyset\Leftrightarrow A\cap f^{−1}(B)=\emptyset.\)
Take \(y\in f(A)\), we know \(y\notin B\). Then \(f^{-1}(y)\in A\) and \(f^{-1}(y)\notin f^{-1}(B)\)
Exercise 9 (⋆)
Let \(f : \mathbb{R} → \mathbb{R}\) be defined by \(f(x) = \frac{2x}{1 + x^2}\).
(a) Is \(f\) injective? surjective?
(b) Show that \(f(\mathbb{R}) = [−1, 1]\).
(c) Show that the restriction \(g : [−1, 1] → [−1, 1], g(x) = f(x)\) is a bijection.
Exercise 10 (⋆)
Let \(f : X → Y\). Show that the following conditions are equivalent:
(a) \(f\) is injective.
(b) For all subsets \(A,B\) of \(X\), we have \(f(A ∩ B) = f(A) ∩ f(B)\).
Exercise 11 (⋆)
Let \(f : \mathbb{Z}× \mathbb{N}^* → \mathbb{Q}, (p, q) ↦ p+ \frac{1}{q}\). Is \(f\) injective, surjective?
\(p_1+\frac{1}{q_1}=p_2+\frac{1}{q_2}\Rightarrow p_1-p_2=\frac{1}{q_2}-\frac{1}{q_1}=n\in\mathbb{Z}\)
Then \(q_2=q_1\), then \(p_1=p_2\)
Take any \(y\in \mathbb{Q}\), if \(y=0.9\), no \(p,q\)