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Definition of Functions

If fx=x2+2x...

Question

If $f (x) = x^{2} + 2 x + 3,$ then find $\frac{f (x + h) - f (x - h)}{h}, h \neq 0$

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Solution

$f (x) = x^{2} + 2 x + 3$

$f (x + h) = {(x + h)}^{2} + 2 (x + h) + 3$

$= x^{2} + 2 x h + h^{2} + 2 x + 2 h + 3$

$f (x - h) = {(x - h)}^{2} + 2 (x - h) + 3$

$= x^{2} - 2 x h + h^{2} + 2 x - 2 h + 3$

Now, $f (x + h) - f (x - h)$

$= x^{2} + 2 x h + h^{2} + 2 x + 2 h + 3 - x^{2} + 2 x h - h^{2} - 2 x + 2 h - 3$

$f (x + h) - f (x - h) = 4 x h + 4 h$

Now, $\frac{f (x + h) - f (x - h)}{h} = \frac{4 x h + 4 h}{h} = 4 x + 4$

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f (x) = 3 x^{2} - 2 x + 5

lim x \to 0 \frac{f (x + h) (- f (x)}{h}

f : R \to R

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