Question

Given that h(x)=x+6 and g(x)= √x −7, find (g+h)(7), if it exists.

A) (g+h)(7)=6
B) The function is undefined.

Answer 1

The combined function (g+h)(7) is defined and equals the sum of both functions evaluated at 7, resulting in √7 + 6.

Step-by-step explanation:

To find (g+h)(7), we need to compute both g(7) and h(7) and then add the results together. The functions given are h(x) = x + 6 and g(x) = √x - 7. Let's evaluate both functions at x = 7:

For h(7), it's straightforward:
h(7) = 7 + 6 = 13.

Next, for g(7), we need to be careful because the square root function is only defined for non-negative numbers:
g(7) = √7 - 7. Since the square root of 7 is a positive real number, this expression is defined, and we can calculate it.

Now we need to combine these:
(g+h)(7) = g(7) + h(7)(g+h)(7) = (√7 - 7) + 13

Finally, simplify the expression:
(g+h)(7) = √7 + 6, which is a real number.

So, the correct answer is The function is defined, and (g+h)(7) equals √7 + 6.