For each of the following real-valued functions on the real numbers \(\mathbb{R}\), indicate whether it is a bijection, a surjection but not a bijection, an injection but not a bijection, or neither an injection nor a surjection.
\(x+2\)
\(2x \)
\(x^2 \)
\(x^2 \) is not a surjection, since negative numbers could not be squares of real numbers. \(x^2 \) also is not an injection, since \((-1)^2=1^2\).
\(x^3 \)
\(\sin x \)
\(\sin x \) is not a surjection, since \(-1 \leq \sin x \leq 1 \). \(\sin x \) is not an injection, since \(\sin 0 = \sin 2\pi \).
\(x \sin x \)
\(x \sin x \) is not an injection, since \(0 \sin 0 = 2\pi \sin 2 \pi \).
\(e^x \)
\(e^x \) is not a surjection, since \(e^x \) is always positive for real values of \(x \).
