Let f ( x )

Given the function $f(x) = x^2 + 3x + 7$, we are tasked with finding $f\left(\frac{x}{2}\right)$.

Step-by-step process:

1. Substitute $\frac{x}{2}$ into the function: $f\left(\frac{x}{2}\right) = \left(\frac{x}{2}\right)^2 + 3\left(\frac{x}{2}\right) + 7$

2. Simplify each term:

   • $\left(\frac{x}{2}\right)^2 = \frac{x^2}{4}$
   • $3\left(\frac{x}{2}\right) = \frac{3x}{2}$

   So the expression becomes: $f\left(\frac{x}{2}\right) = \frac{x^2}{4} + \frac{3x}{2} + 7$

3. Combine the terms: $f\left(\frac{x}{2}\right) = \frac{x^2}{4} + \frac{3x}{2} + 7$ This is the simplified form of $f\left(\frac{x}{2}\right)$.

Would you like any further details or explanations on this?

Here are 5 follow-up questions:

1. What is the value of $f\left(\frac{x}{2}\right)$ when $x = 4$?
2. How does changing the coefficient of $x^2$ in $f(x)$ affect the function?
3. Can you rewrite the expression $f\left(\frac{x}{2}\right)$ in terms of a common denominator?
4. What happens if you apply this method to other functions, such as $g(x) = x^3 - 2x + 5$?
5. What is the domain of the function $f(x) = x^2 + 3x + 7$?

Tip: Always pay attention to distributing correctly when substituting variables in functions, as it's easy to miss details!