Solved on Feb 26, 2024

Find the range of $g(x) = -8^x$ . (Answer in interval notation)

STEP 1

Assumptions
1. The function is $g(x) = -8^x$ .
2. We are looking for the range of the function, which is the set of all possible output values.
3. The domain of the function is all real numbers, as there are no restrictions on the exponent for real number bases.

STEP 2

Understand the behavior of the function $g(x) = -8^x$ .
1. The base of the exponent is $8$ , which is a positive real number.
2. The negative sign in front of $8^x$ indicates that the function will take the opposite sign of the value of $8^x$ .

STEP 3

Analyze the behavior of $8^x$ as $x$ varies.
1. As $x$ approaches $+\infty$ , $8^x$ grows without bound, so $8^x \to +\infty$ .
2. As $x$ approaches $-\infty$ , $8^x$ approaches $0$ , so $8^x \to 0$ .

STEP 4

Apply the negative sign to the behavior of $8^x$ .
1. Since $8^x$ grows without bound as $x \to +\infty$ , $-8^x$ will decrease without bound, so $-8^x \to -\infty$ .
2. Since $8^x$ approaches $0$ as $x \to -\infty$ , $-8^x$ will also approach $0$ , but from the negative side, so $-8^x \to 0$ .

STEP 5

Determine the range of $g(x) = -8^x$ .
1. From the previous steps, we know that $-8^x$ will take on all negative values from $-\infty$ up to but not including $0$ .
2. Therefore, the range of $g(x)$ is all negative real numbers.

STEP 6

Express the range in interval notation.
The range of $g(x) = -8^x$ is $(-\infty, 0)$ .

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Find the range of $g(x) = -8^x$ . (Answer in interval notation)

STEP 1

Assumptions
1. The function is $g(x) = -8^x$ .
2. We are looking for the range of the function, which is the set of all possible output values.
3. The domain of the function is all real numbers, as there are no restrictions on the exponent for real number bases.

STEP 2

Understand the behavior of the function $g(x) = -8^x$ .
1. The base of the exponent is $8$ , which is a positive real number.
2. The negative sign in front of $8^x$ indicates that the function will take the opposite sign of the value of $8^x$ .

STEP 3

Analyze the behavior of $8^x$ as $x$ varies.
1. As $x$ approaches $+\infty$ , $8^x$ grows without bound, so $8^x \to +\infty$ .
2. As $x$ approaches $-\infty$ , $8^x$ approaches $0$ , so $8^x \to 0$ .

STEP 4

Apply the negative sign to the behavior of $8^x$ .
1. Since $8^x$ grows without bound as $x \to +\infty$ , $-8^x$ will decrease without bound, so $-8^x \to -\infty$ .
2. Since $8^x$ approaches $0$ as $x \to -\infty$ , $-8^x$ will also approach $0$ , but from the negative side, so $-8^x \to 0$ .

STEP 5

Determine the range of $g(x) = -8^x$ .
1. From the previous steps, we know that $-8^x$ will take on all negative values from $-\infty$ up to but not including $0$ .
2. Therefore, the range of $g(x)$ is all negative real numbers.

STEP 6

Express the range in interval notation.
The range of $g(x) = -8^x$ is $(-\infty, 0)$ .

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Find the range of g(x)=−8xg(x) = -8^xg(x)=−8x. (Answer in interval notation)

STEP 1

Assumptions1. The function is g(x)=−8xg(x) = -8^xg(x)=−8x.2. We are looking for the range of the function, which is the set of all possible output values.3. The domain of the function is all real numbers, as there are no restrictions on the exponent for real number bases.

STEP 2

Understand the behavior of the function g(x)=−8xg(x) = -8^xg(x)=−8x.1. The base of the exponent is 888, which is a positive real number.2. The negative sign in front of 8x8^x8x indicates that the function will take the opposite sign of the value of 8x8^x8x.

STEP 3

Analyze the behavior of 8x8^x8x as xxx varies.1. As xxx approaches +∞+\infty+∞, 8x8^x8x grows without bound, so 8x→+∞8^x \to +\infty8x→+∞.2. As xxx approaches −∞-\infty−∞, 8x8^x8x approaches 000, so 8x→08^x \to 08x→0.

STEP 4

STEP 5

Determine the range of g(x)=−8xg(x) = -8^xg(x)=−8x.1. From the previous steps, we know that −8x-8^x−8x will take on all negative values from −∞-\infty−∞ up to but not including 000.2. Therefore, the range of g(x)g(x)g(x) is all negative real numbers.

STEP 6

Express the range in interval notation.The range of g(x)=−8xg(x) = -8^xg(x)=−8x is (−∞,0)(-\infty, 0)(−∞,0).

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Find the range of $g(x) = -8^x$ . (Answer in interval notation)

Assumptions
1. The function is $g(x) = -8^x$ .
2. We are looking for the range of the function, which is the set of all possible output values.
3. The domain of the function is all real numbers, as there are no restrictions on the exponent for real number bases.

Understand the behavior of the function $g(x) = -8^x$ .
1. The base of the exponent is $8$ , which is a positive real number.
2. The negative sign in front of $8^x$ indicates that the function will take the opposite sign of the value of $8^x$ .

Analyze the behavior of $8^x$ as $x$ varies.
1. As $x$ approaches $+\infty$ , $8^x$ grows without bound, so $8^x \to +\infty$ .
2. As $x$ approaches $-\infty$ , $8^x$ approaches $0$ , so $8^x \to 0$ .

Determine the range of $g(x) = -8^x$ .
1. From the previous steps, we know that $-8^x$ will take on all negative values from $-\infty$ up to but not including $0$ .
2. Therefore, the range of $g(x)$ is all negative real numbers.

Express the range in interval notation.
The range of $g(x) = -8^x$ is $(-\infty, 0)$ .