Questions: Given the function f(x)=6x+8, evaluate and simplify the expressions below. f(a)= f(a+h)= (f(a+h)-f(a))/h=

Transcript text: Given the function $f(x)=6 x+8$, evaluate and simplify the expressions below. \[ f(a)= \] \[ f(a+h)= \] \[ \frac{f(a+h)-f(a)}{h}= \]

Solution

Solution Steps

To solve the given problem, we need to evaluate the function $ f(x) = 6x + 8 $ at different points and simplify the resulting expressions.

Evaluate $ f(a) $ by substituting $ a $ into the function.
Evaluate $ f(a+h) $ by substituting $ a+h $ into the function.
Compute the difference quotient $ \frac{f(a+h) - f(a)}{h} $ and simplify.

Step 1: Evaluate $ f(a) $

To find $ f(a) $, we substitute $ a = 5 $ into the function $ f(x) = 6x + 8 $: \[ f(a) = 6(5) + 8 = 30 + 8 = 38 \]

Step 2: Evaluate $ f(a+h) $

To find $ f(a+h) $, we substitute $ a = 5 $ and $ h = 2 $ into the function $ f(x) = 6x + 8 $: \[ f(a+h) = f(5+2) = f(7) = 6(7) + 8 = 42 + 8 = 50 \]

Step 3: Compute the Difference Quotient

To find the difference quotient $ \frac{f(a+h) - f(a)}{h} $, we use the values obtained in the previous steps: \[ \frac{f(a+h) - f(a)}{h} = \frac{50 - 38}{2} = \frac{12}{2} = 6.0 \]

Final Answer

\[ f(a) = 6a + 8 \] \[ f(a+h) = 6(a+h) + 8 = 6a + 6h + 8 \] \[ \frac{f(a+h) - f(a)}{h} = \frac{(6a + 6h + 8) - (6a + 8)}{h} = \frac{6h}{h} = 6 \]

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