In function notation, decide whether the missing value is an input or an output

$f\left(4\right)$ asks for the output at input 4

A function assigns an output to each allowed input. For $f\left(x\right)=2x^{2}-3x+1$, $f\left(4\right)$ means replace every x with 4. Define the function once, then enter $f\left(4\right)$ on a second expression line. The result is 21.

The symbol f names the rule. $f\left(4\right)$ does not mean f multiplied by 4. On the graph, the corresponding point is $\left(4,21\right)$.

$\displaystyle f\left(x\right)=2x^{2}-3x+1$

Define the function.

$\displaystyle f\left(4\right)$

Calculate its output: 21.

$f\left(x\right)=10$ asks which inputs produce 10

For $f\left(x\right)=3x-2$, the equation $f\left(x\right)=10$ gives the output and asks you to find an input. Graph f and $y=10$, then select their intersection. At $\left(4,10\right)$, the input is 4.

A quadratic can reach the same output at two different inputs. Check every intersection in the relevant domain. If the question asks for a positive input, a negative intersection may solve the equation but fail that condition.

$\displaystyle f\left(x\right)=3x-2$

Graph the function.

$\displaystyle y=10$

Graph the known output. Read x at the crossing.

Nested functions act from the inside out

Let $f\left(x\right)=x^{2}+1$ and $g\left(x\right)=2x-1$. To find $f\left(g\left(3\right)\right)$, first calculate $g\left(3\right)=5$, then use 5 as the input to f. This gives $f\left(5\right)=26$.

After defining both functions, enter $f\left(g\left(3\right)\right)$ to check the calculation. The order matters: $g\left(f\left(3\right)\right)$ would give 19. For a table of outputs, place $f\left(x_{1}\right)$ in a table heading after defining f.

Read transformations in the input carefully

For $f\left(x-2\right)$, the function receives $x-2$ as its input. If x is 5, the input to f is 3. The graph shifts right by 2 because you must increase x by 2 to reach the same original input.

Graph $f\left(x\right)$ and $f\left(x-2\right)$ on separate lines to compare them. For $f\left(x\right)+2$, the addition acts on the output, so the graph shifts up by 2. The placement of the parentheses determines which change occurs.

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