QuestionIdentify which functions are one-to-one: $f(x)=\frac{x-1}{3x+3}$ , $f(x)=\sqrt{5x+9}$ , $f(x)=\frac{7}{4x^2}$ , $f(x)=\frac{1}{2}x^3$ , $f(x)=3x^4+7x^3$ .

Studdy Solution

STEP 1

Assumptions1. We are given five functions and we need to determine which of them are one-to-one. . A function is one-to-one (also known as injective) if every element of the function's domain maps to a unique element of its range. In other words, no two different elements in the domain of the function have the same image in the range of the function.

STEP 2

To check if a function is one-to-one, we can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one.

STEP 3

Let's start by analyzing the first function $f(x)=\frac{x-1}{3 x+3}$ .

STEP 4

This function can be simplified to $f(x)=\frac{1}{3}-\frac{1}{3x+3}$ .

STEP 5

The graph of this function is a hyperbola, which is not one-to-one because it fails the horizontal line test.

STEP 6

Next, let's analyze the second function $f(x)=\sqrt{5 x+9}$ .

STEP 7

The graph of this function is a square root function shifted to the left by9 units and vertically stretched by a factor of5. This function is one-to-one because it passes the horizontal line test.

STEP 8

Next, let's analyze the third function $f(x)=\frac{7}{4 x^{2}}$ .

STEP 9

The graph of this function is a hyperbola, which is not one-to-one because it fails the horizontal line test.

STEP 10

Next, let's analyze the fourth function $f(x)=\frac{}{2} x^{3}$ .

STEP 11

The graph of this function is a cubic function, which is one-to-one because it passes the horizontal line test.

STEP 12

Finally, let's analyze the fifth function $f(x)= x^{4}+7 x^{}$ .

STEP 13

The graph of this function is a quartic function, which is not one-to-one because it fails the horizontal line test.
Therefore, the one-to-one functions are $f(x)=\sqrt{5 x+9}$ and $f(x)=\frac{}{2} x^{3}$ .

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QuestionIdentify which functions are one-to-one: $f(x)=\frac{x-1}{3x+3}$ , $f(x)=\sqrt{5x+9}$ , $f(x)=\frac{7}{4x^2}$ , $f(x)=\frac{1}{2}x^3$ , $f(x)=3x^4+7x^3$ .

Studdy Solution

STEP 1

STEP 2

To check if a function is one-to-one, we can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one.

STEP 3

Let's start by analyzing the first function $f(x)=\frac{x-1}{3 x+3}$ .

STEP 4

This function can be simplified to $f(x)=\frac{1}{3}-\frac{1}{3x+3}$ .

STEP 5

The graph of this function is a hyperbola, which is not one-to-one because it fails the horizontal line test.

STEP 6

Next, let's analyze the second function $f(x)=\sqrt{5 x+9}$ .

STEP 7

The graph of this function is a square root function shifted to the left by9 units and vertically stretched by a factor of5. This function is one-to-one because it passes the horizontal line test.

STEP 8

Next, let's analyze the third function $f(x)=\frac{7}{4 x^{2}}$ .

STEP 9

The graph of this function is a hyperbola, which is not one-to-one because it fails the horizontal line test.

STEP 10

Next, let's analyze the fourth function $f(x)=\frac{}{2} x^{3}$ .

STEP 11

The graph of this function is a cubic function, which is one-to-one because it passes the horizontal line test.

STEP 12

Finally, let's analyze the fifth function $f(x)= x^{4}+7 x^{}$ .

STEP 13

The graph of this function is a quartic function, which is not one-to-one because it fails the horizontal line test.
Therefore, the one-to-one functions are $f(x)=\sqrt{5 x+9}$ and $f(x)=\frac{}{2} x^{3}$ .

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QuestionIdentify which functions are one-to-one: f(x)=x−13x+3f(x)=\frac{x-1}{3x+3}f(x)=3x+3x−1​, f(x)=5x+9f(x)=\sqrt{5x+9}f(x)=5x+9​, f(x)=74x2f(x)=\frac{7}{4x^2}f(x)=4x27​, f(x)=12x3f(x)=\frac{1}{2}x^3f(x)=21​x3, f(x)=3x4+7x3f(x)=3x^4+7x^3f(x)=3x4+7x3.

Studdy Solution

STEP 1

STEP 2

To check if a function is one-to-one, we can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one.

STEP 3

Let's start by analyzing the first function f(x)=x−13x+3f(x)=\frac{x-1}{3 x+3}f(x)=3x+3x−1​.

STEP 4

This function can be simplified to f(x)=13−13x+3f(x)=\frac{1}{3}-\frac{1}{3x+3}f(x)=31​−3x+31​.

STEP 5

The graph of this function is a hyperbola, which is not one-to-one because it fails the horizontal line test.

STEP 6

Next, let's analyze the second function f(x)=5x+9f(x)=\sqrt{5 x+9}f(x)=5x+9​.

STEP 7

The graph of this function is a square root function shifted to the left by9 units and vertically stretched by a factor of5. This function is one-to-one because it passes the horizontal line test.

STEP 8

Next, let's analyze the third function f(x)=74x2f(x)=\frac{7}{4 x^{2}}f(x)=4x27​.

STEP 9

The graph of this function is a hyperbola, which is not one-to-one because it fails the horizontal line test.

STEP 10

Next, let's analyze the fourth function f(x)=2x3f(x)=\frac{}{2} x^{3}f(x)=2​x3.

STEP 11

The graph of this function is a cubic function, which is one-to-one because it passes the horizontal line test.

STEP 12

Finally, let's analyze the fifth function f(x)=x4+7xf(x)= x^{4}+7 x^{}f(x)=x4+7x.

STEP 13

The graph of this function is a quartic function, which is not one-to-one because it fails the horizontal line test.Therefore, the one-to-one functions are f(x)=5x+9f(x)=\sqrt{5 x+9}f(x)=5x+9​ and f(x)=2x3f(x)=\frac{}{2} x^{3}f(x)=2​x3.

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QuestionIdentify which functions are one-to-one: $f(x)=\frac{x-1}{3x+3}$ , $f(x)=\sqrt{5x+9}$ , $f(x)=\frac{7}{4x^2}$ , $f(x)=\frac{1}{2}x^3$ , $f(x)=3x^4+7x^3$ .

Let's start by analyzing the first function $f(x)=\frac{x-1}{3 x+3}$ .

This function can be simplified to $f(x)=\frac{1}{3}-\frac{1}{3x+3}$ .

Next, let's analyze the second function $f(x)=\sqrt{5 x+9}$ .

Next, let's analyze the third function $f(x)=\frac{7}{4 x^{2}}$ .

Next, let's analyze the fourth function $f(x)=\frac{}{2} x^{3}$ .

Finally, let's analyze the fifth function $f(x)= x^{4}+7 x^{}$ .

The graph of this function is a quartic function, which is not one-to-one because it fails the horizontal line test.
Therefore, the one-to-one functions are $f(x)=\sqrt{5 x+9}$ and $f(x)=\frac{}{2} x^{3}$ .