Steps

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range to domain

Composition

Let $f$ be a function with domain $D_f$.

Let $g$ be a function with domain $D_g$.

Then the composition of f and g, $f \circ g$, is defined as

\[ (f \circ g)(x) = f(g(x)) \]

The value of $g$ becomes a domain number for $f$.

The composition is defined on a subset of the domain of $g$. The composition is defined at those numbers in the domain of $g$ where the value of $g$ is in the domain of $f$.

The values of $g$ are becoming domain numbers for $f$.

The domain of $(f \circ g)(x) = f(g(x))$ is not the domain of $g$.
The domain of $(f \circ g)(x) = f(g(x))$ is a subset of the domain of $g$.
The domain of $(f \circ g)(x) = f(g(x))$ is consists of the domain numbers of $g$, for which the value of $g$ is in the domain of $f$.

Let $X(a) = -\sqrt {3 - a}$ with its natural domain.

Let $Y(b) = \ln (1 + b)$ with its natural domain.

Let $Z(c) = \frac {3}{5 - c}$ with its natural domain.

Is $2$ in the domain of $X$?

Yes No

Is $2$ in the domain of $Y \circ X$?

Yes No

Is $2$ in the domain of $Z \circ X$?

Yes No

Let $X(a) = -\sqrt {3 - a}$ with its natural domain.

Let $Y(b) = \ln (1 + b)$ with its natural domain.

Let $Z(c) = \frac {3}{5 - c}$ with its natural domain.

Is $27$ in the domain of $Y$?

Yes No

Is $27$ in the domain of $X \circ Y$?

Yes No

Is $27$ in the domain of $Z \circ Y$?

Yes No

Let $X(a) = -\sqrt {3 - a}$ with its natural domain.

Let $Y(b) = \ln (1 + b)$ with its natural domain.

Let $Z(c) = \frac {3}{5 - c}$ with its natural domain.

Is $5$ in the domain of $Z$?

Yes No

Is $5$ in the domain of $X \circ Z$?

Yes No

Is $5$ in the domain of $Y \circ Z$?

Yes No

Define $f$ and $g$ graphically as follows:

Is $2$ in the domain of $f \circ g$?

Yes No

Is $2$ in the domain of $g \circ f$?

Yes No

Is $1$ in the domain of $f \circ g$?

Yes No

Is $1$ in the domain of $g \circ f$?

Yes No

Define $f$ and $g$ graphically as follows:

Is $1$ in the domain of $f \circ g$?

Yes No

Is $1$ in the domain of $g \circ f$?

Yes No

Is $-1$ in the domain of $f \circ g$?

Yes No

Is $-1$ in the domain of $g \circ f$?

Yes No

Is $0$ in the domain of $g \circ g$?

Yes No

Is $1$ in the domain of $g \circ g$?

Yes No

Is $2$ in the domain of $g \circ g$?

Yes No

Is $-2$ in the domain of $f \circ g$?

Yes No

Is $-2$ in the domain of $g \circ f$?

Yes No

Is $-2$ in the domain of $g \circ g$?

Yes No

Is $-2$ in the domain of $f \circ f$?

Yes No

Range Values

Let $H(x) = e^x + 20$ and $K(y) = \sqrt {y}$.

Let $C(t) = (K \circ H)(t) = K(H(t))$.

$4$ is in the range of $K$, since $16$ is in the natural domain of $K$ and $K(16) = \sqrt {16} = 4$.

However, $4$ is not in the range of $K \circ H$, because the value of $H$ is never $16$.

Is $1$ in the range of $K$?

Yes No

Is $1$ in the range of $K \circ H$?

Yes No

The range of $K \circ H$ is inside the range of $K$, but it might not be all of the range of $K$.

It depends on how much of the domain of $K$ gets covered by the range of $H$.

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more examples can be found by following this link
More Examples of Composition

2025-07-18 16:24:59