Composite Functions Flashcards

All 7 cards in this deck

What does the composite function $fg(x)$ mean?

Apply $g$ first, then apply $f$ to the result — $f(g(x))$.

What are the steps to evaluate $fg(a)$?

e.g. $fg(1)$ where $f(x)=x+5$ and $g(x)=3x$

1. Work out the inner function: $g(1)=3$
2. Put that answer into $f$: $f(3)=3+5$
3. State the value: $fg(1)=8$

What are the steps to find an expression for $fg(x)$?

e.g. $f(x)=2x-1$, $g(x)=x^2$

1. Write $f$ with $g(x)$ as its input: $f(x^2)$
2. Substitute $g(x)$ for every $x$ in $f$: $2(x^2)-1$
3. Simplify: $fg(x)=2x^2-1$

True or false? $fg(x)$ and $gf(x)$ always give the same expression.

False. The order matters; $fg(x)$ and $gf(x)$ are usually different.

True or false? $fg(x)$ means $f(x)$ multiplied by $g(x)$.

False. It means substitute $g(x)$ into $f$, i.e. $f(g(x))$.

What does $ff(x)$ mean?

e.g. $f(x)=x+3$

Apply $f$ to the output of $f$: $f(f(x))$.
Here $ff(x)=(x+3)+3=x+6$.

What are the steps to find an unknown constant from a condition on a composite function, such as $fg(2)=10$?

e.g. $f(x)=x+a$, $g(x)=3x$

1. Apply the inner function: $g(2)=6$
2. Apply the outer function, keeping the constant: $f(6)=6+a$
3. Set equal to the given value and solve: $6+a=10$, so $a=4$