Question

For each of the following functions determine the image of S = {x ∈ R : 9 ≤ x²}.

1. f : R → R defined by f(x) = |x|.

2. g : R → R⁺ defined by g(x) = e^x.

3. h : R → R defined by h(x) = x − 9

Answer 1

$\displaystyle S = \{x \in \mathcal{R}: 9 \leq x^2\}\\ S = \{x \in \mathcal{R}: x^2 \geq 9\},\,\,\, S = \{x \in \mathcal{R}: x \leq -3,\,\, x \geq 3\} \\ 1.\,\,\, f(x) = |x|, \\ f(-3) = |-3| = 3 \\ f(3) = |3| = 3 \\ \textsf{Image of}\,\, f \,\,\textsf{is}\,\, \{x \in \mathcal{R}: f(x) \geq 3\} \\ 2.\,\,\,g(x) = e^x, \\ g(-3) = e^{-3} \\ g(3) = e^{3} \\ \textsf{Image of}\,\, g \,\,\textsf{is}\,\, \{x \in \mathcal{R}: g(x) \leq e^{-3},\,\, g(x) \geq e^{3}\} \\ 3.\,\,\,h(x) = x - 9, \\ h(-3) = -3 - 9 = -12 \\ h(3) = 3 - 9 = -6 \\ \textsf{Image of}\,\, h \,\,\textsf{is}\,\, \{x \in \mathcal{R}: h(x) \leq -12,\,\, h(x) \geq -6\}$