Math / Algebra

QuestionFor number 10 and 11 graph the function and identify the domain and range. 10) $f(x)=\sqrt{x-3}$ 11) $g(x)=\sqrt{x+4}$
For number 12-13 solve the equation and check your answer. 12) $3 \sqrt[3]{5 x+6}=18$

Studdy Solution

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.
$3 \times 6 = 18$

STEP 25

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

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QuestionFor number 10 and 11 graph the function and identify the domain and range. 10) f(x)=x−3f(x)=\sqrt{x-3}f(x)=x−3​ 11) g(x)=x+4g(x)=\sqrt{x+4}g(x)=x+4​For number 12-13 solve the equation and check your answer. 12) 35x+63=183 \sqrt[3]{5 x+6}=18335x+6​=18

Studdy Solution

STEP 1

Assumptions for Problem 10 and 111. We are given two functions: f(x)=x−3 f(x) = \sqrt{x-3} f(x)=x−3​ and g(x)=x+4 g(x) = \sqrt{x+4} g(x)=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)=x−3 f(x) = \sqrt{x-3} f(x)=x−3​, determine the domain.The expression under the square root, x−3 x-3 x−3, must be non-negative for the function to be defined.x−3≥0 x - 3 \geq 0 x−3≥0

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

Assumptions for Problem 121. We are given the equation 35x+63=18 3 \sqrt[3]{5x+6} = 18 335x+6​=18.2. We need to solve for x x x.3. We need to check the solution.

STEP 11

Isolate the cube root on one side of the equation.5x+63=183 \sqrt[3]{5x+6} = \frac{18}{3} 35x+6​=318​

STEP 12

Simplify the right side of the equation.5x+63=6 \sqrt[3]{5x+6} = 6 35x+6​=6

STEP 13

Cube both sides to eliminate the cube root.5x+6=63 5x + 6 = 6^3 5x+6=63

STEP 14

Calculate 63 6^3 63.63=216 6^3 = 216 63=216

STEP 15

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

STEP 16

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

STEP 17

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

STEP 18

Divide both sides by 5 to solve for x x x.x=2105 x = \frac{210}{5} x=5210​

STEP 19

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

STEP 20

Check the solution by substituting x=42 x = 42 x=42 back into the original equation.35(42)+63=18 3 \sqrt[3]{5(42) + 6} = 18 335(42)+6​=18

STEP 21

Calculate inside the cube root.3210+63=18 3 \sqrt[3]{210 + 6} = 18 33210+6​=18

STEP 22

Simplify inside the cube root.32163=18 3 \sqrt[3]{216} = 18 33216​=18

STEP 23

Calculate the cube root of 216.2163=6 \sqrt[3]{216} = 6 3216​=6

STEP 24

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

STEP 25

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

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QuestionFor number 10 and 11 graph the function and identify the domain and range. 10) $f(x)=\sqrt{x-3}$ 11) $g(x)=\sqrt{x+4}$
For number 12-13 solve the equation and check your answer. 12) $3 \sqrt[3]{5 x+6}=18$

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.

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.
$x - 3 \geq 0$

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

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

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.
$x + 4 \geq 0$

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

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

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.

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.

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.

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

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

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

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

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

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

Calculate the right side.
$5x = 210$

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

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

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

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

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

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

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.