To shift the function \( f(x) = (x + 1)^3 \) to the left by three units, you need to replace \( x \) in the function with \( x + 3 \).
So, the shifted function will be: \[ f(x) = ((x + 3) + 1)^3 = (x + 4)^3 \]
Now, you want to express this in the form of the equation "Half of x equals \((x + \text{blank})^3 + \text{blank}\)".
First, let's define half of \( x \): \[ \frac{1}{2}x \]
Now we need to express this as: \[ \frac{1}{2}x = (x + \text{blank})^3 + \text{blank} \]
With our newly shifted function: \[ \frac{1}{2}x = (x + 4)^3 + \text{blank} \]
From the left side and the cubic function derived earlier, we see that we can set:
- The first blank equals \( 4 \) (from the function \((x + 4)^3\))
- The second blank needs to be the result of that term to balance the equation.
To find the value of the second blank, we need to compute the constant adjustment necessary to modify the form. However, the specific "blank" for balancing will depend on additional information about how you want to relate it back to some target value for a comparison or solution.
Thus, without loss of generalization: \[ \frac{1}{2}x = (x + 4)^3 + b \]
For specific values of the second blank, you may choose any fixed constant or desired comparison.
Here is the full derived equation: \[ \frac{1}{2}x = (x + 4)^3 + b \]
Given that this definition has a variable outcome based on your choice of \( b \), it would typically require more context to finalize. However, if you want to set \( b = -\frac{1}{2}x - (x + 4)^3\), you may simplify or adjust it to integrate constant shifts as needed.