Math Snap

STEP 1

1. We are given two functions: $f(x) = \sqrt{1-x}$ and $g(x) = x^2$.
2. We need to find the composition $(f \circ g)(x)$.
3. We need to determine the domain and range of $(f \circ g)(x)$.

STEP 2

1. Find the expression for $(f \circ g)(x)$.
2. Determine the domain of $(f \circ g)(x)$.
3. Determine the range of $(f \circ g)(x)$.

STEP 3

To find $(f \circ g)(x)$, substitute $g(x) = x^2$ into $f(x)$:
$$(f \circ g)(x) = f(g(x)) = f(x^2) = \sqrt{1 - x^2}$$

STEP 4

Determine the domain of $(f \circ g)(x) = \sqrt{1 - x^2}$.
The expression under the square root, $1 - x^2$, must be non-negative:
$$1 - x^2 \geq 0$$ This implies:
$$1 \geq x^2$$ Taking square roots:
$$-1 \leq x \leq 1$$ Thus, the domain of $(f \circ g)(x)$ is:
$$[-1, 1]$$

Determine the range of $(f \circ g)(x) = \sqrt{1 - x^2}$.
The expression $\sqrt{1 - x^2}$ represents the upper half of a circle with radius 1 centered at the origin. The values of $\sqrt{1 - x^2}$ range from 0 to 1 as $x$ varies from $-1$ to $1$.
Thus, the range of $(f \circ g)(x)$ is:
$$[0, 1]$$ The expression for $(f \circ g)(x)$ is $\sqrt{1 - x^2}$, with domain $[-1, 1]$ and range $[0, 1]$.