Functions Flashcards

All 20 cards in this deck

In function notation, what does $f(3)$ mean?

The output of the function $f$ when the input $x$ is $3$.
e.g. if $f(x)=2x+1$ then $f(3)=7$.

How do you evaluate $f(-3)$ for a given function?

e.g. $f(x)=2x^2$

Substitute $-3$ in place of every $x$, keeping the negative in brackets.
$f(-3)=2\times(-3)^2=18$

What are the steps to solve $f(x)=k$ for $x$?

e.g. solve $f(x)=11$ where $f(x)=2x+3$

1. Write the equation: $2x+3=11$
2. Solve using inverse operations: $2x=8$
3. Give $x$: $x=4$

For which value of $x$ is a function of the form $f(x)=\frac{a}{x-b}$ undefined?

e.g. $f(x)=\frac{5}{x-2}$

$x=b$, because the denominator would be $0$.
Here $x=2$.

True or false? $f(x+2)$ means the same as $f(x)+2$.

False. $f(x+2)$ means substitute $x+2$ as the input; $f(x)+2$ means add $2$ to the output.

True or false? In $f(x)=3x^2$, writing $f(4)$ means $f$ multiplied by $4$.

False. $f(4)$ means the value of the function when the input is $4$, not a multiplication.

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 is the inverse function $f^{-1}(x)$?

The function that reverses $f$: it takes an output of $f$ back to its input.

What are the steps to find $f^{-1}(x)$?

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

1. Write $y=f(x)$: $y=2x-1$
2. Rearrange to make $x$ the subject: $x=\frac{y+1}{2}$
3. Replace $y$ with $x$: $f^{-1}(x)=\frac{x+1}{2}$

What are two ways to find $f^{-1}(k)$ for a given number $k$?

e.g. $f^{-1}(7)$ where $f(x)=2x+1$

Find $f^{-1}(x)$ and substitute $k$, or solve $f(x)=k$.
Here $2x+1=7$ gives $f^{-1}(7)=3$.

True or false? $f^{-1}(x)$ means $\frac{1}{f(x)}$.

False. $f^{-1}$ is the inverse function, not the reciprocal.

True or false? $ff^{-1}(x)=x$ for all values of $x$ in the domain.

True. Applying a function then its inverse returns the original input.

How do you find $h^{-1}(x)$ when $h(x)=fg(x)$?

e.g. $f(x)=x^3$, $g(x)=2x+3$

Form the composite first, then invert it.
$h(x)=(2x+3)^3$, so $h^{-1}(x)=\frac{\sqrt[3]{x}-3}{2}$

What are the steps to find an unknown constant from a condition such as $f^{-1}(13)=g(2)$?

e.g. $f(x)=3x+1$, $g(x)=ax$

1. Work out the known side: $f^{-1}(13)=\frac{13-1}{3}=4$
2. Write the other side in terms of the constant: $g(2)=2a$
3. Equate and solve: $2a=4$, so $a=2$

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 $a=4$