If $^{8\ f(x)+1\ =\ x^2}$ , what is $f(1−2x)$$?$

Given the equation $ 8f(x) + 1 = x^2 $, we want to determine $ f(1 - 2x) $.

### Step 1: Solve for $ f(x) $
First, solve the given equation for $ f(x) $:
\[
8f(x) = x^2 - 1
\]
\[
f(x) = \frac{x^2 - 1}{8}
\]

### Step 2: Find $ f(1 - 2x) $
Now, substitute $ 1 - 2x $ into the expression for $ f(x) $:
\[
f(1 - 2x) = \frac{(1 - 2x)^2 - 1}{8}
\]

### Step 3: Simplify the expression
Expand $ (1 - 2x)^2 $:
\[
(1 - 2x)^2 = 1 - 4x + 4x^2
\]
So, we have:
\[
f(1 - 2x) = \frac{1 - 4x + 4x^2 - 1}{8}
\]
Simplify the expression:
\[
f(1 - 2x) = \frac{4x^2 - 4x}{8}
\]
\[
f(1 - 2x) = \frac{4x(x - 1)}{8}
\]
\[
f(1 - 2x) = \frac{x(x - 1)}{2}
\]

### Final Answer
\[
f(1 - 2x) = \frac{x(x - 1)}{2}
\]

Would you like any further details or have any questions about this solution?

Here are some related questions:
1. What is $ f(2) $ for the given function?
2. How would you find $ f(2x) $ for the same function $ f(x) $?
3. What is the value of $ f(0) $?
4. How can the expression for $ f(x) $ be used to determine its roots?
5. What is the derivative of $ f(x) $ with respect to $ x $?

**Tip:** When substituting into a function, always expand and simplify the expression carefully to avoid errors.

Discover how to solve for f(1 - 2x) in the equation 8f(x) + 1 = x^2. This step-by-step guide covers key algebraic concepts, including function manipulation and quadratic expressions. Perfect for high school students in Grades 9-11.

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