Algebraically

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replacing

Algebraically, we just replace.

Algebraically

Let $Out(x) = 3x^3 + \sin (4x) - \frac {3}{2-x}$ with its natural or implied domain.
Let $In(t) = \frac {5t-7}{t^2-8}$ with its natural or implied domain.

Algebraically, composition is accomplished by replacing all occurences of the variable with the entire formula for the other function.

\[ (Out \circ In)(k) = Out(In(k)) = 3 {\frac {5k-7}{k^2-8}}^3 + \sin (4 \frac {5k-7}{k^2-8}) - \frac {3}{2-\frac {5t-7}{k^2-8}} \]

Obviously, parentheses are vital here.

\[ (Out \circ In)(k) = Out(In(k)) = 3 \left ( \frac {5k-7}{k^2-8} \right )^3 + \sin \left ( 4 \left ( \frac {5k-7}{k^2-8} \right ) \right ) - \frac {3}{2 - \left ( \frac {5k-7}{k^2-8} \right )} \]

We can always make two compositions from two functions.

\[ (In \circ Out)(u) = In(Out(u)) = \frac {5 \left ( 3u^3 + \sin (4u) - \frac {3}{2-u} \right )-7}{\left ( 3u^3 + \sin (4u) - \frac {3}{2-u} \right )^2-8} \]

Composition

Let $f(x) = 5x^2 - 4x + 2$ with its natural or implied domain.
Let $g(t) = \frac {3}{6-t}$ with its natural or implied domain.

\[ (f \circ g)(y) = f(g(y)) = 5 \left ( \answer {\frac {3}{6-y}} \right )^2 - 4 \left ( \answer {\frac {3}{6-y}} \right ) + 2 \]

\[ (g \circ f)(w) = g(f(w)) = \frac {3}{6 - \left ( \answer {5w^2 - 4w + 2} \right )} \]

Composition

Let $f(x) = 3x^2 - x + 5$ with its natural or implied domain.
Let $g(t) = \frac {4}{7-t}$ with its natural or implied domain.

Thinking about $(g \circ f)(w) = g(f(w))$...

There are two numbers where $f(x) = 7$. One of them is $1$, the other is $\answer {\frac {-2}{3}}$.

\[ (g \circ f)(w) = g(f(w)) = \frac {4}{7 - (5w^2 - 4w + 2)} \]

In the $g \circ f$ composition, $f$ is no longer allowed to equal $\answer {7}$.

Select all real numbers which cannot be in the domain of $g \circ f$.

$7$ $0$ $1$ $\frac {2}{3}$ $-1$ $\frac {-2}{3}$

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

2025-01-07 00:25:23