Question

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=x2−4x+2ƒ(x)=$$x^2-4x+2$$

Accepted Answer

Start by finding $$ f(x+h)  by $$substituting $$ x+h $$ into the function $$ f(x) = x^2 - 4x + 2 . $$This means replacing every $$ x  in $$the expression with $$ x+h $$, so write $$ f(x+h) = (x+h)^2 - 4(x+h) + 2 . $$Next, expand the squared term and distribute the $$ -4 $$ across $$ (x+h) . $$Recall that $$ (x+h)^2 = x^2 + 2xh + h^2 $$ and $$ -4(x+h) = -4x - 4h . So$$, rewrite $$ f(x+h)  as$$ $$ x^2 + 2xh + h^2 - 4x - 4h + 2 . $$Now, find $$ f(x+h) - f(x)  by $$subtracting the original function $$ f(x) = x^2 - 4x + 2 $$ from the expression you found for $$ f(x+h) . $$Write this as $$ (x^2 + 2xh + h^2 - 4x - 4h + 2) - (x^2 - 4x + 2) . $$Simplify the expression $$ f(x+h) - f(x)  by $$canceling out like terms. Notice that $$ x^2 $$, $$ -4x $$, and $$ +2 $$ will cancel out, leaving you with the terms involving $$ h . $$Finally, find $$ \frac{f(x+h) - f(x)}{h}  by $$dividing the simplified expression from the previous step by $$ h . $$This will give you the difference quotient, which is a key concept in understanding rates of change and the derivative.

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4.
$ƒ (x) = x^{2} - 4 x + 2$

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Key Concepts

Function Notation and Evaluation

Difference of Function Values

Difference Quotient

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=x2−4x+2ƒ(x)=x^2-4x+2