Solved on Jan 17, 2024

Solve the exponential equation $y=9^{x}$ for $x$ in terms of $y$ .

STEP 1

Assumptions
1. We are given the function $y = 9^{x}$ .
2. We want to understand the properties of this function.

STEP 2

Identify the type of function.
The function $y = 9^{x}$ is an exponential function because the variable $x$ is in the exponent.

STEP 3

Determine the base of the exponential function.
The base of the exponential function is 9, which is a positive real number greater than 1.

STEP 4

Analyze the growth of the function.
Since the base is greater than 1, the function $y = 9^{x}$ represents exponential growth. As $x$ increases, $y$ increases rapidly.

STEP 5

Consider the domain of the function.
The domain of $y = 9^{x}$ is all real numbers, $-\infty < x < \infty$ , because you can raise 9 to any real power.

STEP 6

Consider the range of the function.
The range of $y = 9^{x}$ is all positive real numbers, $0 < y < \infty$ , because an exponential function with a base greater than 1 will never be zero or negative.

STEP 7

Identify the y-intercept.
The y-intercept occurs when $x = 0$ . Plugging $x = 0$ into the function, we get $y = 9^{0}$ .

STEP 8

Calculate the y-intercept.
Since any non-zero number to the power of 0 is 1, we have $y = 9^{0} = 1$ . Therefore, the y-intercept is $(0, 1)$ .

STEP 9

Determine the behavior as $x$ approaches infinity.
As $x$ approaches infinity, $y = 9^{x}$ grows without bound, so $y$ approaches infinity.

STEP 10

Determine the behavior as $x$ approaches negative infinity.
As $x$ approaches negative infinity, $y = 9^{x}$ approaches 0 because the function represents a very small positive number that gets closer and closer to 0.

STEP 11

Consider the horizontal asymptote.
The horizontal asymptote of $y = 9^{x}$ is the line $y = 0$ because the function approaches 0 as $x$ approaches negative infinity.

STEP 12

Plot the function.
To visualize the function, you can plot several points and draw the curve that represents the exponential growth. Make sure to include the y-intercept and the horizontal asymptote in your graph.

Was this helpful?

Solve the exponential equation $y=9^{x}$ for $x$ in terms of $y$ .

STEP 1

Assumptions
1. We are given the function $y = 9^{x}$ .
2. We want to understand the properties of this function.

STEP 2

Identify the type of function.
The function $y = 9^{x}$ is an exponential function because the variable $x$ is in the exponent.

STEP 3

Determine the base of the exponential function.
The base of the exponential function is 9, which is a positive real number greater than 1.

STEP 4

Analyze the growth of the function.
Since the base is greater than 1, the function $y = 9^{x}$ represents exponential growth. As $x$ increases, $y$ increases rapidly.

STEP 5

Consider the domain of the function.
The domain of $y = 9^{x}$ is all real numbers, $-\infty < x < \infty$ , because you can raise 9 to any real power.

STEP 6

Consider the range of the function.
The range of $y = 9^{x}$ is all positive real numbers, $0 < y < \infty$ , because an exponential function with a base greater than 1 will never be zero or negative.

STEP 7

Identify the y-intercept.
The y-intercept occurs when $x = 0$ . Plugging $x = 0$ into the function, we get $y = 9^{0}$ .

STEP 8

Calculate the y-intercept.
Since any non-zero number to the power of 0 is 1, we have $y = 9^{0} = 1$ . Therefore, the y-intercept is $(0, 1)$ .

STEP 9

Determine the behavior as $x$ approaches infinity.
As $x$ approaches infinity, $y = 9^{x}$ grows without bound, so $y$ approaches infinity.

STEP 10

Determine the behavior as $x$ approaches negative infinity.
As $x$ approaches negative infinity, $y = 9^{x}$ approaches 0 because the function represents a very small positive number that gets closer and closer to 0.

STEP 11

Consider the horizontal asymptote.
The horizontal asymptote of $y = 9^{x}$ is the line $y = 0$ because the function approaches 0 as $x$ approaches negative infinity.

STEP 12

Plot the function.
To visualize the function, you can plot several points and draw the curve that represents the exponential growth. Make sure to include the y-intercept and the horizontal asymptote in your graph.

Start learning now

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

Solve the exponential equation y=9xy=9^{x}y=9x for xxx in terms of yyy.

STEP 1

Assumptions1. We are given the function y=9x y = 9^{x} y=9x.2. We want to understand the properties of this function.

STEP 2

Identify the type of function.The function y=9x y = 9^{x} y=9x is an exponential function because the variable x x x is in the exponent.

STEP 3

Determine the base of the exponential function.The base of the exponential function is 9, which is a positive real number greater than 1.

STEP 4

Analyze the growth of the function.Since the base is greater than 1, the function y=9x y = 9^{x} y=9x represents exponential growth. As x x x increases, y y y increases rapidly.

STEP 5

Consider the domain of the function.The domain of y=9x y = 9^{x} y=9x is all real numbers, −∞<x<∞ -\infty < x < \infty −∞<x<∞, because you can raise 9 to any real power.

STEP 6

Consider the range of the function.The range of y=9x y = 9^{x} y=9x is all positive real numbers, 0<y<∞ 0 < y < \infty 0<y<∞, because an exponential function with a base greater than 1 will never be zero or negative.

STEP 7

Identify the y-intercept.The y-intercept occurs when x=0 x = 0 x=0. Plugging x=0 x = 0 x=0 into the function, we get y=90 y = 9^{0} y=90.

STEP 8

Calculate the y-intercept.Since any non-zero number to the power of 0 is 1, we have y=90=1 y = 9^{0} = 1 y=90=1. Therefore, the y-intercept is (0,1) (0, 1) (0,1).

STEP 9

Determine the behavior as x x x approaches infinity.As x x x approaches infinity, y=9x y = 9^{x} y=9x grows without bound, so y y y approaches infinity.

STEP 10

Determine the behavior as x x x approaches negative infinity.As x x x approaches negative infinity, y=9x y = 9^{x} y=9x approaches 0 because the function represents a very small positive number that gets closer and closer to 0.

STEP 11

Consider the horizontal asymptote.The horizontal asymptote of y=9x y = 9^{x} y=9x is the line y=0 y = 0 y=0 because the function approaches 0 as x x x approaches negative infinity.

STEP 12

Plot the function.To visualize the function, you can plot several points and draw the curve that represents the exponential growth. Make sure to include the y-intercept and the horizontal asymptote in your graph.

Start learning now

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

Solve the exponential equation $y=9^{x}$ for $x$ in terms of $y$ .

Assumptions
1. We are given the function $y = 9^{x}$ .
2. We want to understand the properties of this function.

Identify the type of function.
The function $y = 9^{x}$ is an exponential function because the variable $x$ is in the exponent.

Determine the base of the exponential function.
The base of the exponential function is 9, which is a positive real number greater than 1.

Analyze the growth of the function.
Since the base is greater than 1, the function $y = 9^{x}$ represents exponential growth. As $x$ increases, $y$ increases rapidly.

Consider the domain of the function.
The domain of $y = 9^{x}$ is all real numbers, $-\infty < x < \infty$ , because you can raise 9 to any real power.

Consider the range of the function.
The range of $y = 9^{x}$ is all positive real numbers, $0 < y < \infty$ , because an exponential function with a base greater than 1 will never be zero or negative.

Identify the y-intercept.
The y-intercept occurs when $x = 0$ . Plugging $x = 0$ into the function, we get $y = 9^{0}$ .

Calculate the y-intercept.
Since any non-zero number to the power of 0 is 1, we have $y = 9^{0} = 1$ . Therefore, the y-intercept is $(0, 1)$ .

Determine the behavior as $x$ approaches infinity.
As $x$ approaches infinity, $y = 9^{x}$ grows without bound, so $y$ approaches infinity.

Determine the behavior as $x$ approaches negative infinity.
As $x$ approaches negative infinity, $y = 9^{x}$ approaches 0 because the function represents a very small positive number that gets closer and closer to 0.

Consider the horizontal asymptote.
The horizontal asymptote of $y = 9^{x}$ is the line $y = 0$ because the function approaches 0 as $x$ approaches negative infinity.

Plot the function.
To visualize the function, you can plot several points and draw the curve that represents the exponential growth. Make sure to include the y-intercept and the horizontal asymptote in your graph.