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Given the function f(x)=4x+6, evaluate and simplify the expressions below. See special instructions on how to enter your answers. f(a)= {:[f(a+h)=],[(f(a+h)-f(a))/(h)=]:}

Given the function f(x)=4x+6 f(x)=4 x+6 , evaluate and simplify the expressions below. See special instructions on how to enter your answers.\newlinef(a)= f(a)= \newlinef(a+h)=f(a+h)−f(a)h= \begin{array}{l} f(a+h)= \\ \frac{f(a+h)-f(a)}{h}= \end{array}

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Full solution

Q. Given the function f(x)=4x+6 f(x)=4 x+6 , evaluate and simplify the expressions below. See special instructions on how to enter your answers.\newlinef(a)= f(a)= \newlinef(a+h)=f(a+h)−f(a)h= \begin{array}{l} f(a+h)= \\ \frac{f(a+h)-f(a)}{h}= \end{array}
  1. Substitute and Find f(a)f(a): To find f(a)f(a), substitute xx with aa in the function f(x)f(x).\newlinef(a)=4a+6f(a) = 4a + 6
  2. Substitute and Find f(a+h)f(a+h): Now, let's find f(a+h)f(a+h). Substitute xx with (a+h)(a+h) in the function f(x)f(x).\newlinef(a+h)=4(a+h)+6f(a+h) = 4(a+h) + 6
  3. Simplify f(a+h)f(a+h): Simplify f(a+h)f(a+h).f(a+h)=4a+4h+6f(a+h) = 4a + 4h + 6
  4. Calculate f(a+h)−f(a)f(a+h)-f(a): Next, we need to find f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}. First, calculate f(a+h)−f(a)f(a+h)-f(a).\newlinef(a+h)−f(a)=(4a+4h+6)−(4a+6)f(a+h)-f(a) = (4a + 4h + 6) - (4a + 6)
  5. Simplify f(a+h)−f(a)f(a+h)-f(a): Simplify the expression f(a+h)−f(a)f(a+h)-f(a).f(a+h)−f(a)=4a+4h+6−4a−6f(a+h)-f(a) = 4a + 4h + 6 - 4a - 6
  6. Cancel Like Terms: Cancel out the like terms in f(a+h)−f(a)f(a+h)-f(a). \newlinef(a+h)−f(a)=4hf(a+h)-f(a) = 4h
  7. Divide by h: Now, divide by h to find (f(a+h)−f(a))/h(f(a+h)-f(a))/h.(f(a+h)−f(a))/h=4hh(f(a+h)-f(a))/h = \frac{4h}{h}
  8. Simplify Result: Simplify the expression (f(a+h)−f(a))/h(f(a+h)-f(a))/h.(f(a+h)−f(a))/h=4(f(a+h)-f(a))/h = 4

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