Math / Algebra

QuestionAnswer all of the following questions. (1) What is the domain of $f(x)=\frac{3}{x+4}$ ? Explain how you reached your conclusion. (2) Consider the graph of $f(x)=\frac{3}{x+4}$ shown below. What do you think is the range of $f(x)$ ?

Studdy Solution

STEP 1

1. The function $f(x) = \frac{3}{x+4}$ is a rational function.
2. The domain of a function is the set of all possible input values (x-values) that do not make the function undefined.
3. The range of a function is the set of all possible output values (y-values) that the function can take.

STEP 2

1. Determine the domain of $f(x)$ .
2. Determine the range of $f(x)$ .

STEP 3

To find the domain of $f(x) = \frac{3}{x+4}$ , identify the values of $x$ that make the denominator zero, as these will make the function undefined.

STEP 4

Set the denominator equal to zero: $x + 4 = 0$ .

STEP 5

Solve for $x$ : $x = -4$ .

STEP 6

The domain of $f(x)$ is all real numbers except $x = -4$ . In interval notation, this is $(-\infty, -4) \cup (-4, \infty)$ .

STEP 7

To determine the range of $f(x) = \frac{3}{x+4}$ , consider the behavior of the function as $x$ approaches the vertical asymptote $x = -4$ and the horizontal asymptote $y = 0$ .

STEP 8

As $x \to -4^+$ or $x \to -4^-$ , $f(x)$ approaches $\pm \infty$ , indicating that $y$ can take any real value except the horizontal asymptote value.

STEP 9

The horizontal asymptote at $y = 0$ suggests that $f(x)$ never actually reaches $y = 0$ .

STEP 10

Thus, the range of $f(x)$ is all real numbers except $y = 0$ . In interval notation, this is $(-\infty, 0) \cup (0, \infty)$ .
The domain of $f(x)$ is $(-\infty, -4) \cup (-4, \infty)$ .
The range of $f(x)$ is $(-\infty, 0) \cup (0, \infty)$ .

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QuestionAnswer all of the following questions. (1) What is the domain of $f(x)=\frac{3}{x+4}$ ? Explain how you reached your conclusion. (2) Consider the graph of $f(x)=\frac{3}{x+4}$ shown below. What do you think is the range of $f(x)$ ?

Studdy Solution

STEP 1

1. The function $f(x) = \frac{3}{x+4}$ is a rational function.
2. The domain of a function is the set of all possible input values (x-values) that do not make the function undefined.
3. The range of a function is the set of all possible output values (y-values) that the function can take.

STEP 2

1. Determine the domain of $f(x)$ .
2. Determine the range of $f(x)$ .

STEP 3

To find the domain of $f(x) = \frac{3}{x+4}$ , identify the values of $x$ that make the denominator zero, as these will make the function undefined.

STEP 4

Set the denominator equal to zero: $x + 4 = 0$ .

STEP 5

Solve for $x$ : $x = -4$ .

STEP 6

The domain of $f(x)$ is all real numbers except $x = -4$ . In interval notation, this is $(-\infty, -4) \cup (-4, \infty)$ .

STEP 7

To determine the range of $f(x) = \frac{3}{x+4}$ , consider the behavior of the function as $x$ approaches the vertical asymptote $x = -4$ and the horizontal asymptote $y = 0$ .

STEP 8

As $x \to -4^+$ or $x \to -4^-$ , $f(x)$ approaches $\pm \infty$ , indicating that $y$ can take any real value except the horizontal asymptote value.

STEP 9

The horizontal asymptote at $y = 0$ suggests that $f(x)$ never actually reaches $y = 0$ .

STEP 10

Thus, the range of $f(x)$ is all real numbers except $y = 0$ . In interval notation, this is $(-\infty, 0) \cup (0, \infty)$ .
The domain of $f(x)$ is $(-\infty, -4) \cup (-4, \infty)$ .
The range of $f(x)$ is $(-\infty, 0) \cup (0, \infty)$ .

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QuestionAnswer all of the following questions. (1) What is the domain of f(x)=3x+4f(x)=\frac{3}{x+4}f(x)=x+43​ ? Explain how you reached your conclusion. (2) Consider the graph of f(x)=3x+4f(x)=\frac{3}{x+4}f(x)=x+43​ shown below. What do you think is the range of f(x)f(x)f(x) ?

Studdy Solution

STEP 1

STEP 2

1. Determine the domain of f(x) f(x) f(x).2. Determine the range of f(x) f(x) f(x).

STEP 3

To find the domain of f(x)=3x+4 f(x) = \frac{3}{x+4} f(x)=x+43​, identify the values of x x x that make the denominator zero, as these will make the function undefined.

STEP 4

Set the denominator equal to zero: x+4=0 x + 4 = 0 x+4=0.

STEP 5

Solve for x x x: x=−4 x = -4 x=−4.

STEP 6

The domain of f(x) f(x) f(x) is all real numbers except x=−4 x = -4 x=−4. In interval notation, this is (−∞,−4)∪(−4,∞) (-\infty, -4) \cup (-4, \infty) (−∞,−4)∪(−4,∞).

STEP 7

To determine the range of f(x)=3x+4 f(x) = \frac{3}{x+4} f(x)=x+43​, consider the behavior of the function as x x x approaches the vertical asymptote x=−4 x = -4 x=−4 and the horizontal asymptote y=0 y = 0 y=0.

STEP 8

As x→−4+ x \to -4^+ x→−4+ or x→−4− x \to -4^- x→−4−, f(x) f(x) f(x) approaches ±∞ \pm \infty ±∞, indicating that y y y can take any real value except the horizontal asymptote value.

STEP 9

The horizontal asymptote at y=0 y = 0 y=0 suggests that f(x) f(x) f(x) never actually reaches y=0 y = 0 y=0.

STEP 10

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QuestionAnswer all of the following questions. (1) What is the domain of $f(x)=\frac{3}{x+4}$ ? Explain how you reached your conclusion. (2) Consider the graph of $f(x)=\frac{3}{x+4}$ shown below. What do you think is the range of $f(x)$ ?

1. Determine the domain of $f(x)$ .
2. Determine the range of $f(x)$ .

To find the domain of $f(x) = \frac{3}{x+4}$ , identify the values of $x$ that make the denominator zero, as these will make the function undefined.

Set the denominator equal to zero: $x + 4 = 0$ .

Solve for $x$ : $x = -4$ .

The domain of $f(x)$ is all real numbers except $x = -4$ . In interval notation, this is $(-\infty, -4) \cup (-4, \infty)$ .

To determine the range of $f(x) = \frac{3}{x+4}$ , consider the behavior of the function as $x$ approaches the vertical asymptote $x = -4$ and the horizontal asymptote $y = 0$ .

As $x \to -4^+$ or $x \to -4^-$ , $f(x)$ approaches $\pm \infty$ , indicating that $y$ can take any real value except the horizontal asymptote value.

The horizontal asymptote at $y = 0$ suggests that $f(x)$ never actually reaches $y = 0$ .