Math Snap

STEP 1

What is this asking?
We're plugging $\pi$ into $\cos(x)$, and then plugging that result into $g(x) = \sqrt{10 + x}$.
Watch out!
Make sure to evaluate the functions from the inside out!
Also, remember what $\cos(\pi)$ equals.

STEP 2

1. Evaluate $f(\pi)$
2. Evaluate $g(f(\pi))$

STEP 3

Alright, so we start with our function $f(x) = \cos(x)$.
We want to find $f(\pi)$, which just means plugging in $\pi$ for $x$.
So, we get $f(\pi) = \cos(\pi)$.

STEP 4

Now, remember the unit circle!
At $\pi$ radians, the x-coordinate is -1, which is precisely the value of $\cos(\pi)$.
So, $f(\pi) = \cos(\pi) = \mathbf{-1}$.
Awesome!

STEP 5

We know that $g(x) = \sqrt{10 + x}$ and we just found that $f(\pi) = \mathbf{-1}$.
This means $g(f(\pi))$ is the same as $g(\mathbf{-1})$.

STEP 6

Let's plug in $\mathbf{-1}$ for $x$ in $g(x)$.
We get $g(\mathbf{-1}) = \sqrt{10 + (\mathbf{-1})}$.

STEP 7

Inside the square root, we have $10 + (\mathbf{-1}) = 10 - 1 = \mathbf{9}$.
So, our expression becomes $\sqrt{\mathbf{9}}$.

STEP 8

Finally, the square root of $\mathbf{9}$ is $\mathbf{3}$!
So, $g(f(\pi)) = g(\mathbf{-1}) = \sqrt{10 + (\mathbf{-1})} = \sqrt{\mathbf{9}} = \mathbf{3}$.
We did it!

The value of $g(f(\pi))$ is $\mathbf{3}$.