Math Snap

STEP 1

Assumptions for Problem 10 and 11
1. We are given two functions: $f(x) = \sqrt{x-3}$ and $g(x) = \sqrt{x+4}$.
2. We need to graph each function.
3. We need to identify the domain and range of each function.

STEP 2

For $f(x) = \sqrt{x-3}$, determine the domain.
The expression under the square root, $x-3$, must be non-negative for the function to be defined.
$$x - 3 \geq 0$$

STEP 3

Solve the inequality for the domain of $f(x)$.
$$x \geq 3$$ Thus, the domain of $f(x)$ is $[3, \infty)$.

STEP 4

Identify the range of $f(x) = \sqrt{x-3}$.
Since the square root function produces non-negative outputs, the range is:
$$[0, \infty)$$

STEP 5

For $g(x) = \sqrt{x+4}$, determine the domain.
The expression under the square root, $x+4$, must be non-negative for the function to be defined.
$$x + 4 \geq 0$$

STEP 6

Solve the inequality for the domain of $g(x)$.
$$x \geq -4$$ Thus, the domain of $g(x)$ is $[-4, \infty)$.

STEP 7

Identify the range of $g(x) = \sqrt{x+4}$.
Since the square root function produces non-negative outputs, the range is:
$$[0, \infty)$$

STEP 8

Graph $f(x) = \sqrt{x-3}$.
1. The graph starts at the point $(3, 0)$.
2. The graph is a half-parabola opening to the right.
3. As $x$ increases, $f(x)$ increases.

STEP 9

Graph $g(x) = \sqrt{x+4}$.
1. The graph starts at the point $(-4, 0)$.
2. The graph is a half-parabola opening to the right.
3. As $x$ increases, $g(x)$ increases.

STEP 10

Assumptions for Problem 12
1. We are given the equation $3 \sqrt[3]{5x+6} = 18$.
2. We need to solve for $x$.
3. We need to check the solution.

STEP 11

Isolate the cube root on one side of the equation.
$$\sqrt[3]{5x+6} = \frac{18}{3}$$

STEP 12

Simplify the right side of the equation.
$$\sqrt[3]{5x+6} = 6$$

STEP 13

Cube both sides to eliminate the cube root.
$$5x + 6 = 6^3$$

STEP 14

Calculate $6^3$.
$$6^3 = 216$$

STEP 15

Substitute back into the equation.
$$5x + 6 = 216$$

STEP 16

Subtract 6 from both sides to solve for $5x$.
$$5x = 216 - 6$$

STEP 17

Calculate the right side.
$$5x = 210$$

STEP 18

Divide both sides by 5 to solve for $x$.
$$x = \frac{210}{5}$$

STEP 19

Calculate the value of $x$.
$$x = 42$$

STEP 20

Check the solution by substituting $x = 42$ back into the original equation.
$$3 \sqrt[3]{5(42) + 6} = 18$$

STEP 21

Calculate inside the cube root.
$$3 \sqrt[3]{210 + 6} = 18$$

STEP 22

Simplify inside the cube root.
$$3 \sqrt[3]{216} = 18$$

STEP 23

Calculate the cube root of 216.
$$\sqrt[3]{216} = 6$$

STEP 24

Multiply by 3 to check the left side of the original equation.
$$3 \times 6 = 18$$

Since both sides of the equation are equal, the solution $x = 42$ is verified.