First slide

Introduction to real valued functions

Question

Let the function
$f (x) = 3 x^{2} - 4 x + 8 \log (1 + |x|)$ be defined on the interval [0, 1]. The even extension of f(x) of the interval [-1, 1] is

Moderate

A

$3 x^{2} - 4 |x| + 8 \log (1 + |x|)$
B

$3 x^{2} - 4 x + 8 \log (1 + |x|)$
C

$3 x^{2} + 4 x - 8 \log (1 + |x|)$
D

$3 x^{2} - 4 x - 8 \log (1 + |x|)$

Solution

$f (x) = - f (x)$ $x \in [0, 1]$
$- f (- x)$ $x \in [- 1, 0]$
$f (x) = 3 x^{2} - 4 x + 8 \log (1 + |x|) x \in [0, 1]$
$f (x) = 3 x^{2} + 4 x + 8 \log (1 + |x|) x \in [- 1, 0]$
$∴ f (x) = 3 x^{2} - 4 |x| + 8 \log (1 + |x|) x \in [- 1, 1]$

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