Exercise 5
If \(f(x) = x (x - 1)\), then \( f(-x) \) equals to:
-
\(f(x)\)
-
\(-f(x)\)
-
\(f(1 - x)\)
-
\(f(x + 1)\)
-
\(f(x - 1)\)
Try to solve it before checking the answer.
\(f(x + 1)\)
Indeed:
\[
\begin{aligned}
f(x) &= (-x) ( (-x) – 1 )
\\[.5em]
&=
(-x) (-x -1)
\\[.5em]
&=
(-x) (-1) (x + 1)
\\[.5em]
&=
x (x + 1)
\end{aligned}
\]
\begin{aligned}
f(x) &= (-x) ( (-x) – 1 )
\\[.5em]
&=
(-x) (-x -1)
\\[.5em]
&=
(-x) (-1) (x + 1)
\\[.5em]
&=
x (x + 1)
\end{aligned}
\]
On the other hand,
\[
\begin{aligned}
f(x + 1) &= (x + 1) ( (x + 1) – 1 )
\\[.5em]
&= (x + 1) x
\\[.5em]
&= x (x + 1)
\end{aligned}
\]
\begin{aligned}
f(x + 1) &= (x + 1) ( (x + 1) – 1 )
\\[.5em]
&= (x + 1) x
\\[.5em]
&= x (x + 1)
\end{aligned}
\]
Then:
\[
\boldsymbol{
f(-x) = f(x + 1)
}
\]
\boldsymbol{
f(-x) = f(x + 1)
}
\]