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1.4 Composition of functions (1) Given the following functions $f(x)$ and $g(x)$, find the natural domain of the two possible compositions, i.e. $D_{fog}$ and $D_{gof}$: (a) $f(x) = \ln x$, $g(x) = \sqrt{x+3}$ (c) $f(x) = \sqrt{-x}$, $g(x) = x^4 + x^2 + 6$ (b) $f(x) = 2^x$, $g(x) = x^2 + 3x + 1$ (d) $f(x) = \frac{x+2}{x-3}$, $g(x) = 3^x$ (2) Consider the following functions: $f(x) = \frac{1}{x}$, $g(x) = x^2 + 1$, $h(x) = 2x^2 - 3x + 4$, $w(x) = x^2$. Write down expressions for the following compositions and determine the natural domain D of the resulting function: (a) $h(h(x))$ (d) $f(\ln(h(x))$ (b) $f(\sin(w(x)))$ (c) $\ln(h(x))$ (e) $f(w(\sqrt{x+1}))$ (f) $f(g(h(x))) $

          1.4 Composition of functions
(1) Given the following functions $f(x)$ and $g(x)$, find the natural domain of the two possible
compositions, i.e. $D_{fog}$ and $D_{gof}$:
(a) $f(x) = \ln x$, $g(x) = \sqrt{x+3}$
(c) $f(x) = \sqrt{-x}$, $g(x) = x^4 + x^2 + 6$
(b) $f(x) = 2^x$, $g(x) = x^2 + 3x + 1$
(d) $f(x) = \frac{x+2}{x-3}$, $g(x) = 3^x$
(2) Consider the following functions: $f(x) = \frac{1}{x}$, $g(x) = x^2 + 1$, $h(x) = 2x^2 - 3x + 4$, $w(x) = x^2$.
Write down expressions for the following compositions and determine the natural domain
D of the resulting function:
(a) $h(h(x))$
(d) $f(\ln(h(x))$
(b) $f(\sin(w(x)))$ 
(c) $\ln(h(x))$
(e) $f(w(\sqrt{x+1}))$
(f) $f(g(h(x))) $

1.4 Composition of functions
(1) Given the following functions f(x) and g(x), find the natural domain of the two possible
compositions, i.e. Dfog and Dgof:
(a) f(x) = ln x, g(x) = √(x+3)
(c) f(x) = √(-x), g(x) = x^4 + x^2 + 6
(b) f(x) = 2^x, g(x) = x^2 + 3x + 1
(d) f(x) = (x+2)/(x-3), g(x) = 3^x
(2) Consider the following functions: f(x) = (1)/(x), g(x) = x^2 + 1, h(x) = 2x^2 - 3x + 4, w(x) = x^2.
Write down expressions for the following compositions and determine the natural domain
D of the resulting function:
(a) h(h(x))
(d) f(ln(h(x))
(b) f(sin(w(x)))
(c) ln(h(x))
(e) f(w(√(x+1)))
(f) f(g(h(x)))

Added by Kristi A.

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