Final answer:
The correct value of h(k(x)) when h(x) = 5x^2 - 1 and k(x) = 5x + 1 for x > 0, and following the appropriate substitution and algebraic expansion, is not listed in the provided options. Our calculations yield 125x^2 + 50x + 4, which does not match any of the given choices, suggesting there may be a typo in the question or the provided options.
Step-by-step explanation:
To find the value of h(k(x)) for x > 0 when h(x) = 5x^2 - 1 and k(x) = 5x + 1, we need to substitute k(x) into h(x). This means wherever there is an x in h(x), we replace it with k(x). Let's do that step by step:
- First, write down the function h(x) as given: h(x) = 5x^2 - 1.
- Next, substitute k(x) for x in the function h(x): h(k(x)) = 5(k(x))^2 - 1.
- Now, substitute the actual expression for k(x) into the new function: h(k(x)) = 5(5x + 1)^2 - 1.
- Expand the square: h(k(x)) = 5(25x^2 + 10x + 1) - 1.
- Multiply through by 5: h(k(x)) = 125x^2 + 50x + 5 - 1.
- Simplify the expression: h(k(x)) = 125x^2 + 50x + 4.
- However, we don't see this expression in the options provided. It seems there might be a typo in the options or in our calculations. Assuming the typo is in the options and the correct option should include the term + 50x instead of + 5x or + 10x, then none of the provided options would be correct.
If this is not the case, and we assume that the options given are correct, we need to re-check our calculations:
- Ensuring that we correctly expand (5x + 1)^2 gives us 25x^2 + 10x + 1.
- Then, multiplying through by 5: h(k(x)) = 125x^2 + 50x + 5.
- After subtracting 1: h(k(x)) = 125x^2 + 50x + 4.
It appears that none of the options A), B), C), or D) match our result. The correct value of h(k(x)), based on the correct calculations, is not listed among the options provided.