Consider the relation $y=3x+2$, which is a function.
The $y$-values are determined from the $x$-values, so we say ‘$y$ is a function of $x$, abbreviated to $y=f(x)$.
So, the rule $y=3x+2$ can also be written as follows.
$$ \large \begin{align} f: \mapsto \ &3x+2 \\
&\text{or} \\
f(x)= \ &3x+2 \\
&\text{or} \\
y= \ &3x+2 \end{align} $$
A function $f$ such that $x$ is converted into $3x+2$.
Example 1
If $f(x)=4x-5$, find $f(2)$.
\( \begin{align} \displaystyle
f(2) &= 4 \times 2-5 \\
&= 3
\end{align} \)
Example 2
If $f(x)=x^2-5x+1$, find $f(-1)$.
\( \begin{align} \displaystyle
f(-1) &= (-1)^2-5(-1)+1 \\
&= 7
\end{align} \)
Example 3
If $f(x)=x^2-3x+2$, find $f(x+1)$.
\( \begin{align} \displaystyle
f(x+1) &= (x+1)^2-3(x+1)+2 \\
&= x^2+2x=1-3x-3+2 \\
&= x^2+x
\end{align} \)
Example 4
Given $f(x)=ax+b$, $f(1)=7$ and $f(2)=11$, find $a$ and $b$.
\( \begin{align} \displaystyle
f(1) &= 7 \\
a \times 1 +b &= 7 \\
a+b &= 7 \cdots (1) \\
f(2) &= 11 \\
a \times 2 + b &= 11 \\
2a+b &= 11 \cdots (2) \\
(2) &- (1) \\
(2a+b)-(a+b) &= 11-7 \\
a &= 4 \\
4 + b &= 7 \cdots (1) \\
b &= 3
\end{align} \)
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