To find $f(g(x))$, we will substitute $g(x)$ into $f(x)$.

Given:

• $f(x) = 2x^2 + 1$
• $g(x) = 7x - 9$

Step-by-step solution:

1. Start with $f(g(x))$. This means we substitute $g(x)$ (which is $7x - 9$) into $f(x)$: $f(g(x)) = f(7x - 9)$

2. Now, substitute $7x - 9$ into the function $f(x) = 2x^2 + 1$: $f(7x - 9) = 2(7x - 9)^2 + 1$

3. Expand $(7x - 9)^2$: $(7x - 9)^2 = (7x - 9)(7x - 9) = 49x^2 - 126x + 81$

4. Now substitute this back into the function: $f(7x - 9) = 2(49x^2 - 126x + 81) + 1$

5. Distribute the 2: $2(49x^2 - 126x + 81) = 98x^2 - 252x + 162$

6. Add 1: $f(g(x)) = 98x^2 - 252x + 162 + 1 = 98x^2 - 252x + 163$

Thus, the final result is: $f(g(x)) = 98x^2 - 252x + 163$

Would you like any further explanation or clarification?

5 Related Questions:

1. How do you find the inverse of a composite function like $f(g(x))$?
2. What happens if you compute $g(f(x))$ instead of $f(g(x))$?
3. How do you find the domain of the composite function $f(g(x))$?
4. Can you explain how to expand and simplify binomials like $(7x - 9)^2$?
5. How do you determine whether a composite function is one-to-one or onto?

Tip:

When composing functions, always carefully substitute the second function into the first and handle the algebra step-by-step to avoid mistakes in simplification.