Questions: Consider the function (f(x)=4 x^2-2 x-7) Use it to find (f(3), f(-2)), and (f(a+5)). (f(3)=23) (f(-2)=13) (f(a+5)=4 a^2+38 a+83)

Transcript text: Consider the function $f(x)=4 x^{2}-2 x-7$ Use it to find $f(3), f(-2)$, and $f(a+5)$. \[ \begin{array}{l} f(3)=23 \\ f(-2)=13 \\ f(a+5)=4 a^{2}+38 a+83 \end{array} \]

Solution

Solution Steps

To find the values of the function $ f(x) = 4x^2 - 2x - 7 $ at specific points, substitute the given values into the function. For $ f(3) $, substitute $ x = 3 $. For $ f(-2) $, substitute $ x = -2 $. For $ f(a+5) $, substitute $ x = a+5 $ and simplify the expression.

Step 1: Evaluate $ f(3) $

To find $ f(3) $, substitute $ x = 3 $ into the function $ f(x) = 4x^2 - 2x - 7 $.

\[ f(3) = 4(3)^2 - 2(3) - 7 = 4 \times 9 - 6 - 7 = 36 - 6 - 7 = 23 \]

Step 2: Evaluate $ f(-2) $

To find $ f(-2) $, substitute $ x = -2 $ into the function.

\[ f(-2) = 4(-2)^2 - 2(-2) - 7 = 4 \times 4 + 4 - 7 = 16 + 4 - 7 = 13 \]

Step 3: Evaluate $ f(a+5) $

To find $ f(a+5) $, substitute $ x = a+5 $ into the function and simplify.

\[ f(a+5) = 4(a+5)^2 - 2(a+5) - 7 \]

Expanding the expression:

\[ = 4(a^2 + 10a + 25) - 2a - 10 - 7 \]

\[ = 4a^2 + 40a + 100 - 2a - 10 - 7 \]