Math / Algebra

QuestionWrite the domain and the range of the function as an inequality, using set notation, and using interval notation. Also describe the end behavior of the function or explain why there is no end behavior.
5. The graph of the quadratic function $f(x)=x^{2}+2$ is shown.
6. The graph of the exponential function $f(x)=3^{x}$ is shown.

Studdy Solution

STEP 1

1. For the quadratic function $f(x) = x^2 + 2$ , we need to determine the domain and range.
2. For the exponential function $f(x) = 3^x$ , we need to determine the domain and range.
3. We need to describe the end behavior of both functions.

STEP 2

1. Determine the domain and range of the quadratic function $f(x) = x^2 + 2$ .
2. Determine the domain and range of the exponential function $f(x) = 3^x$ .
3. Describe the end behavior of the quadratic function.
4. Describe the end behavior of the exponential function.

STEP 3

Determine the domain of the quadratic function $f(x) = x^2 + 2$ .
The domain of any quadratic function is all real numbers, since $x^2$ is defined for every real number $x$ .
$\text{Domain:} \quad (-\infty, \infty)$

STEP 4

Determine the range of the quadratic function $f(x) = x^2 + 2$ .
The minimum value of $x^2$ is $0$ , which occurs at $x=0$ . Thus, the minimum value of $f(x)$ is $2$ .
$\text{Range:} \quad [2, \infty)$

STEP 5

Determine the domain of the exponential function $f(x) = 3^x$ .
The domain of an exponential function is all real numbers, since $3^x$ is defined for every real number $x$ .
$\text{Domain:} \quad (-\infty, \infty)$

STEP 6

Determine the range of the exponential function $f(x) = 3^x$ .
The exponential function $3^x$ is always positive and never equals zero. Thus, the range is all positive real numbers.
$\text{Range:} \quad (0, \infty)$

STEP 7

Describe the end behavior of the quadratic function $f(x) = x^2 + 2$ .
As $x \to \infty$ , $f(x) = x^2 + 2 \to \infty$ . As $x \to -\infty$ , $f(x) = x^2 + 2 \to \infty$ .
$\text{End behavior:} \quad \lim_{x \to \infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty$

STEP 8

Describe the end behavior of the exponential function $f(x) = 3^x$ .
As $x \to \infty$ , $f(x) = 3^x \to \infty$ . As $x \to -\infty$ , $f(x) = 3^x \to 0$ .
$\text{End behavior:} \quad \lim_{x \to \infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = 0$

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Studdy Solution

STEP 1

1. For the quadratic function $f(x) = x^2 + 2$ , we need to determine the domain and range.
2. For the exponential function $f(x) = 3^x$ , we need to determine the domain and range.
3. We need to describe the end behavior of both functions.

STEP 2

1. Determine the domain and range of the quadratic function $f(x) = x^2 + 2$ .
2. Determine the domain and range of the exponential function $f(x) = 3^x$ .
3. Describe the end behavior of the quadratic function.
4. Describe the end behavior of the exponential function.

STEP 3

Determine the domain of the quadratic function $f(x) = x^2 + 2$ .
The domain of any quadratic function is all real numbers, since $x^2$ is defined for every real number $x$ .
$\text{Domain:} \quad (-\infty, \infty)$

STEP 4

Determine the range of the quadratic function $f(x) = x^2 + 2$ .
The minimum value of $x^2$ is $0$ , which occurs at $x=0$ . Thus, the minimum value of $f(x)$ is $2$ .
$\text{Range:} \quad [2, \infty)$

STEP 5

Determine the domain of the exponential function $f(x) = 3^x$ .
The domain of an exponential function is all real numbers, since $3^x$ is defined for every real number $x$ .
$\text{Domain:} \quad (-\infty, \infty)$

STEP 6

Determine the range of the exponential function $f(x) = 3^x$ .
The exponential function $3^x$ is always positive and never equals zero. Thus, the range is all positive real numbers.
$\text{Range:} \quad (0, \infty)$

STEP 7

Describe the end behavior of the quadratic function $f(x) = x^2 + 2$ .
As $x \to \infty$ , $f(x) = x^2 + 2 \to \infty$ . As $x \to -\infty$ , $f(x) = x^2 + 2 \to \infty$ .
$\text{End behavior:} \quad \lim_{x \to \infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty$

STEP 8

Describe the end behavior of the exponential function $f(x) = 3^x$ .
As $x \to \infty$ , $f(x) = 3^x \to \infty$ . As $x \to -\infty$ , $f(x) = 3^x \to 0$ .
$\text{End behavior:} \quad \lim_{x \to \infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = 0$

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Studdy Solution

STEP 1

1. For the quadratic function f(x)=x2+2f(x) = x^2 + 2f(x)=x2+2, we need to determine the domain and range.2. For the exponential function f(x)=3xf(x) = 3^xf(x)=3x, we need to determine the domain and range.3. We need to describe the end behavior of both functions.

STEP 2

1. Determine the domain and range of the quadratic function f(x)=x2+2f(x) = x^2 + 2f(x)=x2+2.2. Determine the domain and range of the exponential function f(x)=3xf(x) = 3^xf(x)=3x.3. Describe the end behavior of the quadratic function.4. Describe the end behavior of the exponential function.

STEP 3

Determine the domain of the quadratic function f(x)=x2+2f(x) = x^2 + 2f(x)=x2+2.The domain of any quadratic function is all real numbers, since x2x^2x2 is defined for every real number xxx.Domain:(−∞,∞) \text{Domain:} \quad (-\infty, \infty) Domain:(−∞,∞)

STEP 4

Determine the range of the quadratic function f(x)=x2+2f(x) = x^2 + 2f(x)=x2+2.The minimum value of x2x^2x2 is 000, which occurs at x=0x=0x=0. Thus, the minimum value of f(x)f(x)f(x) is 222.Range:[2,∞) \text{Range:} \quad [2, \infty) Range:[2,∞)

STEP 5

Determine the domain of the exponential function f(x)=3xf(x) = 3^xf(x)=3x.The domain of an exponential function is all real numbers, since 3x3^x3x is defined for every real number xxx.Domain:(−∞,∞) \text{Domain:} \quad (-\infty, \infty) Domain:(−∞,∞)

STEP 6

Determine the range of the exponential function f(x)=3xf(x) = 3^xf(x)=3x.The exponential function 3x3^x3x is always positive and never equals zero. Thus, the range is all positive real numbers.Range:(0,∞) \text{Range:} \quad (0, \infty) Range:(0,∞)

STEP 7

STEP 8

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1. For the quadratic function $f(x) = x^2 + 2$ , we need to determine the domain and range.
2. For the exponential function $f(x) = 3^x$ , we need to determine the domain and range.
3. We need to describe the end behavior of both functions.

1. Determine the domain and range of the quadratic function $f(x) = x^2 + 2$ .
2. Determine the domain and range of the exponential function $f(x) = 3^x$ .
3. Describe the end behavior of the quadratic function.
4. Describe the end behavior of the exponential function.

Determine the domain of the quadratic function $f(x) = x^2 + 2$ .
The domain of any quadratic function is all real numbers, since $x^2$ is defined for every real number $x$ .
$\text{Domain:} \quad (-\infty, \infty)$

Determine the range of the quadratic function $f(x) = x^2 + 2$ .
The minimum value of $x^2$ is $0$ , which occurs at $x=0$ . Thus, the minimum value of $f(x)$ is $2$ .
$\text{Range:} \quad [2, \infty)$

Determine the domain of the exponential function $f(x) = 3^x$ .
The domain of an exponential function is all real numbers, since $3^x$ is defined for every real number $x$ .
$\text{Domain:} \quad (-\infty, \infty)$

Determine the range of the exponential function $f(x) = 3^x$ .
The exponential function $3^x$ is always positive and never equals zero. Thus, the range is all positive real numbers.
$\text{Range:} \quad (0, \infty)$