With function notation you can ask two opposite questions. “What is $f(2)$?” gives an input and asks for the output. “Solve $f(x) = 2$” gives an output and asks for the input. The second question is an equation, and you solve it with the equation skills you already have.
Por qué funciona
$f(x) = 2$ says “the output is $2$.” Writing the rule in place of $f(x)$ turns it into an ordinary equation in $x$. On the graph, you are looking for every point at height $2$: draw the horizontal line $y = 2$ and read the $x$-coordinates where it meets the graph. A function can reach the same output from several inputs, so there may be more than one answer, or none.
Idea clave
To solve $f(x) = k$, set the rule equal to $k$ and solve for $x$. Every solution is an input whose output is $k$; check each by computing $f$ of it.
Worked examples
Ejemplo resuelto: A linear function
If $f(x) = 4x - 7$, solve $f(x) = 9$.
- Set the rule equal to the output: $4x - 7 = 9$.
- Add $7$: $4x = 16$. Divide by $4$: $x = 4$.
- Check: $f(4) = 16 - 7 = 9$.
Respuesta: $x = 4$
Ejemplo resuelto: Two inputs, one output
If $g(x) = x^2 - 4$, solve $g(x) = 5$.
- Set the rule equal to $5$: $x^2 - 4 = 5$, so $x^2 = 9$.
- Both $3$ and $-3$ square to $9$.
Respuesta: $x = 3$ or $x = -3$
Error común
Mixing up $f(2)$ and the solution of $f(x) = 2$. With $f(x) = 4x - 7$: $f(2) = 1$ is an output, while $f(x) = 2$ gives the input $x = \frac{9}{4}$. Ask yourself which one you were handed: if the number sits inside the parentheses it is an input; if $f(x)$ is set equal to it, it is an output.