Math / Algebra

QuestionGraph the function and its inverse using a graphing calculator. Use an inverse drawing feature, if available. Find the domain and the range of $f$ and of $f^{-1}$ . $f(x)=(4 x+12)^{3}$

Studdy Solution

STEP 1

1. The function $f(x) = (4x + 12)^3$ is given.
2. We need to graph the function and its inverse.
3. We need to find the domain and range of both $f$ and $f^{-1}$ .

STEP 2

1. Graph the function $f(x)$ .
2. Determine the inverse function $f^{-1}(x)$ .
3. Graph the inverse function $f^{-1}(x)$ .
4. Find the domain and range of $f(x)$ .
5. Find the domain and range of $f^{-1}(x)$ .

STEP 3

To graph $f(x) = (4x + 12)^3$ , input the function into a graphing calculator.
Observe the shape of the graph, which is a cubic function that has been horizontally shifted and vertically scaled.

STEP 4

To find the inverse, start by setting $y = (4x + 12)^3$ .
Solve for $x$ in terms of $y$ : $y^{1/3} = 4x + 12$ $4x = y^{1/3} - 12$ $x = \frac{y^{1/3} - 12}{4}$
Thus, the inverse function is: $f^{-1}(x) = \frac{x^{1/3} - 12}{4}$

STEP 5

To graph $f^{-1}(x) = \frac{x^{1/3} - 12}{4}$ , input the inverse function into the graphing calculator.
If available, use the inverse drawing feature to verify the graph.

STEP 6

The domain of $f(x) = (4x + 12)^3$ is all real numbers, $\mathbb{R}$ , because any real number can be input into a cubic function.
The range of $f(x)$ is also all real numbers, $\mathbb{R}$ , because a cubic function can output any real number.

STEP 7

The domain of $f^{-1}(x) = \frac{x^{1/3} - 12}{4}$ is all real numbers, $\mathbb{R}$ , because any real number can be input into the cube root function.
The range of $f^{-1}(x)$ is also all real numbers, $\mathbb{R}$ , because the cube root function can output any real number.
The domain and range of $f(x)$ are both $\mathbb{R}$ .
The domain and range of $f^{-1}(x)$ are both $\mathbb{R}$ .

Was this helpful?

QuestionGraph the function and its inverse using a graphing calculator. Use an inverse drawing feature, if available. Find the domain and the range of $f$ and of $f^{-1}$ . $f(x)=(4 x+12)^{3}$

Studdy Solution

STEP 1

1. The function $f(x) = (4x + 12)^3$ is given.
2. We need to graph the function and its inverse.
3. We need to find the domain and range of both $f$ and $f^{-1}$ .

STEP 2

1. Graph the function $f(x)$ .
2. Determine the inverse function $f^{-1}(x)$ .
3. Graph the inverse function $f^{-1}(x)$ .
4. Find the domain and range of $f(x)$ .
5. Find the domain and range of $f^{-1}(x)$ .

STEP 3

To graph $f(x) = (4x + 12)^3$ , input the function into a graphing calculator.
Observe the shape of the graph, which is a cubic function that has been horizontally shifted and vertically scaled.

STEP 4

To find the inverse, start by setting $y = (4x + 12)^3$ .
Solve for $x$ in terms of $y$ : $y^{1/3} = 4x + 12$ $4x = y^{1/3} - 12$ $x = \frac{y^{1/3} - 12}{4}$
Thus, the inverse function is: $f^{-1}(x) = \frac{x^{1/3} - 12}{4}$

STEP 5

To graph $f^{-1}(x) = \frac{x^{1/3} - 12}{4}$ , input the inverse function into the graphing calculator.
If available, use the inverse drawing feature to verify the graph.

STEP 6

The domain of $f(x) = (4x + 12)^3$ is all real numbers, $\mathbb{R}$ , because any real number can be input into a cubic function.
The range of $f(x)$ is also all real numbers, $\mathbb{R}$ , because a cubic function can output any real number.

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionGraph the function and its inverse using a graphing calculator. Use an inverse drawing feature, if available. Find the domain and the range of fff and of f−1f^{-1}f−1. f(x)=(4x+12)3f(x)=(4 x+12)^{3}f(x)=(4x+12)3

Studdy Solution

STEP 1

1. The function f(x)=(4x+12)3 f(x) = (4x + 12)^3 f(x)=(4x+12)3 is given.2. We need to graph the function and its inverse.3. We need to find the domain and range of both f f f and f−1 f^{-1} f−1.

STEP 2

1. Graph the function f(x) f(x) f(x).2. Determine the inverse function f−1(x) f^{-1}(x) f−1(x).3. Graph the inverse function f−1(x) f^{-1}(x) f−1(x).4. Find the domain and range of f(x) f(x) f(x).5. Find the domain and range of f−1(x) f^{-1}(x) f−1(x).

STEP 3

To graph f(x)=(4x+12)3 f(x) = (4x + 12)^3 f(x)=(4x+12)3, input the function into a graphing calculator.Observe the shape of the graph, which is a cubic function that has been horizontally shifted and vertically scaled.

STEP 4

STEP 5

To graph f−1(x)=x1/3−124 f^{-1}(x) = \frac{x^{1/3} - 12}{4} f−1(x)=4x1/3−12​, input the inverse function into the graphing calculator.If available, use the inverse drawing feature to verify the graph.

STEP 6

The domain of f(x)=(4x+12)3 f(x) = (4x + 12)^3 f(x)=(4x+12)3 is all real numbers, R \mathbb{R} R, because any real number can be input into a cubic function.The range of f(x) f(x) f(x) is also all real numbers, R \mathbb{R} R, because a cubic function can output any real number.

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionGraph the function and its inverse using a graphing calculator. Use an inverse drawing feature, if available. Find the domain and the range of $f$ and of $f^{-1}$ . $f(x)=(4 x+12)^{3}$

1. The function $f(x) = (4x + 12)^3$ is given.
2. We need to graph the function and its inverse.
3. We need to find the domain and range of both $f$ and $f^{-1}$ .

1. Graph the function $f(x)$ .
2. Determine the inverse function $f^{-1}(x)$ .
3. Graph the inverse function $f^{-1}(x)$ .
4. Find the domain and range of $f(x)$ .
5. Find the domain and range of $f^{-1}(x)$ .

To graph $f(x) = (4x + 12)^3$ , input the function into a graphing calculator.
Observe the shape of the graph, which is a cubic function that has been horizontally shifted and vertically scaled.

To graph $f^{-1}(x) = \frac{x^{1/3} - 12}{4}$ , input the inverse function into the graphing calculator.
If available, use the inverse drawing feature to verify the graph.

The domain of $f(x) = (4x + 12)^3$ is all real numbers, $\mathbb{R}$ , because any real number can be input into a cubic function.
The range of $f(x)$ is also all real numbers, $\mathbb{R}$ , because a cubic function can output any real number.