Questions: If f(x)=3x^2-2x+9 find f(a-2).

Transcript text: If $f(x)=3 x^{2}-2 x+9$ find $f(a-2)$.

Solution

Solution Steps

To find $ f(a-2) $ for the given function $ f(x) = 3x^2 - 2x + 9 $, we need to substitute $ a-2 $ in place of $ x $ in the function and simplify the resulting expression.

Step 1: Substitute $ a-2 $ into the function

Given the function $ f(x) = 3x^2 - 2x + 9 $, we need to find $ f(a-2) $. Substitute $ x = a-2 $ into the function: \[ f(a-2) = 3(a-2)^2 - 2(a-2) + 9 \]

Step 2: Expand and simplify the expression

First, expand $ (a-2)^2 $: \[ (a-2)^2 = a^2 - 4a + 4 \] Now substitute this back into the function: \[ f(a-2) = 3(a^2 - 4a + 4) - 2(a-2) + 9 \] Distribute the constants: \[ f(a-2) = 3a^2 - 12a + 12 - 2a + 4 + 9 \]

Step 3: Combine like terms

Combine the like terms to simplify the expression: \[ f(a-2) = 3a^2 - 14a + 25 \]

Step 4: Evaluate the function at $ a = 2 $

Substitute $ a = 2 $ into the simplified expression: \[ f(2-2) = 3(2)^2 - 14(2) + 25 \] Calculate the values: \[ f(0) = 3(4) - 28 + 25 = 12 - 28 + 25 = 9 \]

Final Answer

$\boxed{f(a-2) = 3a^2 - 14a + 25}$

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