Function notation f(x) and substitution

If $f(x)=2x-1$, then $f(-3)$ means putting $-3$ in for $x$: $2\cdot(-3)-1=-7$. Substitute negative numbers inside parentheses.

Without parentheses, $-4^2$ is $-16$, a different value.

The definition of the derivative also substitutes expressions like $f(a+h)$. Replace $x$ with the whole of $a+h$: if $f(x)=x^2$, then $f(a+h)=(a+h)^2$.

Examples

1. 01What is f(2)?$\displaystyle f(x)=x^3$AnswerHide

   Answer$\displaystyle 8$

   Steps

   1. Just put $2$ where the $x$ is
   2. $\displaystyle f(2)$$\displaystyle = (2)^{3}$$\displaystyle = 8$
2. 02What is f(1)?$\displaystyle f(x)=2x+3$AnswerHide

   Answer$\displaystyle 5$

   Steps

   1. Just put $1$ where the $x$ is
   2. $\displaystyle f(1)$$\displaystyle = 2(1)+3$$\displaystyle = 5$
3. 03What is f(2)?$\displaystyle f(x)=3x+3$AnswerHide

   Answer$\displaystyle 9$

   Steps

   1. Just put $2$ where the $x$ is
   2. $\displaystyle f(2)$$\displaystyle = 3(2)+3$$\displaystyle = 9$
4. 04What is f(-3)?$\displaystyle f(x)=2x-1$AnswerHide

   Answer$\displaystyle -7$

   Steps

   1. Just put $-3$ where the $x$ is
   2. $\displaystyle f(-3)$$\displaystyle = 2(-3)-1$$\displaystyle = -7$
5. 05What is f(-4)?$\displaystyle f(x)=x^2$AnswerHide

   Answer$\displaystyle 16$

   Steps

   1. Just put $-4$ where the $x$ is
   2. $\displaystyle f(-4)$$\displaystyle = (-4)^{2}$$\displaystyle = 16$
6. 06What is f(2)?$\displaystyle f(x)=2x-2$AnswerHide

   Answer$\displaystyle 2$

   Steps

   1. Just put $2$ where the $x$ is
   2. $\displaystyle f(2)$$\displaystyle = 2(2)-2$$\displaystyle = 2$

FAQ

Q1Is $f(x)$ the product of $f$ and $x$?AnswerHide

A

No. $f(x)$ is notation for the value of the function $f$ at $x$, and $f(2)$ is the value when $x=2$.

Q2If $f(x)=x^2$, isn't $f(-4)=-16$?AnswerHide

A

It is $16$. Put $-4$ in with its parentheses: $(-4)^2=16$. $-4^2$ means $-(4^2)=-16$, a different expression.