Find and simplify $$ \frac{(a+h)-f(a)}{h},(h \neq 0) $$ for the following function. $$ f(x)=4 x^{2}-2 x+7 $$ -2

Question

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To simplify the expression

$$
\frac{(a+h)-f(a)}{h}
$$

for the function

$$
f(x) = 4x^2 - 2x + 7,
$$

we will follow these steps:

1. **Calculate $$ f(a) $$**: Substitute $$ a $$ into the function $$ f(x) $$.
2. **Substitute $$ f(a) $$ into the expression**: Replace $$ f(a) $$ in the expression.
3. **Simplify the expression**: Perform algebraic simplifications.

### Step 1: Calculate $$ f(a) $$

Substituting $$ a $$ into the function:

$$
f(a) = 4a^2 - 2a + 7.
$$

### Step 2: Substitute $$ f(a) $$ into the expression

Now, we substitute $$ f(a) $$ into the expression:

$$
\frac{(a+h) - f(a)}{h} = \frac{(a+h) - (4a^2 - 2a + 7)}{h}.
$$

### Step 3: Simplify the expression

Now, we simplify the numerator:

$$
a + h - (4a^2 - 2a + 7) = a + h - 4a^2 + 2a - 7 = -4a^2 + 3a + h - 7.
$$

Thus, the expression becomes:

$$
\frac{-4a^2 + 3a + h - 7}{h}.
$$

Now, we can separate the terms in the numerator:

$$
\frac{-4a^2 + 3a - 7}{h} + \frac{h}{h} = \frac{-4a^2 + 3a - 7}{h} + 1.
$$

So, the final simplified expression is:

$$
\frac{-4a^2 + 3a - 7}{h} + 1.
$$

This is the simplified form of the given expression.
The simplified expression is $$ \frac{-4a^2 + 3a - 7}{h} + 1 $$.

Find and simplify

for the following function.

-2

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Find and simplify for the following function. -2