Graph the function. State its domain and range. y=-2^x+3 |...

Name: Problem 1, Chapter 1: Calculus: Graphical, Numerical, Algebraic
Uploaded: 2021-08-27T00:24:37-08:00
Duration: 4 min 40 s
Channel: Kian Manafi
Description: Graph the function. State its domain and range. y=-2^x+3

Key Concepts

Range

The range of a function is the set of all possible output values (y-values) the function can produce. For transformed exponential functions, the range is often determined by any horizontal asymptotes created through vertical shifts and reflections, influencing the values the function can approach but not necessarily reach.

Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For exponential functions, and even after transformations like reflections and vertical shifts, the domain usually remains all real numbers.

Exponential Functions

Exponential functions are mathematical functions in the form f(x) = a^x, where the base a is a positive constant not equal to 1. These functions are characterized by very rapid increases or decreases and exhibit continuous growth or decay, making them important in various natural and economic models.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a given function. For exponential functions, applying a reflection (via a negative multiplier) flips the graph vertically, and adding a constant moves it up or down. These modifications change the function's appearance and properties such as the range, without altering the domain.

Transcript

00:01 Okay, so we're going to be stating the domain range of this function and then graphing it.

00:04 So the domain is all of the x values that give us a defined y value or that when we plug in that x value, we actually get a value back for y that is a real number.

00:16 And so in this case, 2 to the x, we can actually plug in any real number for x and we'll get a real value back for y.

00:23 So our domain here is going to be from negative infinity to infinity.

00:27 And the reason for that is because the function, or the exponential function, 2 to the x, there are no x values in the real numbers that if we plugged it in, we would get an undefined function back.

00:37 And so because of that, our domain is all real numbers.

00:41 The range is going to be all the y values that our x values can produce.

00:45 So for all of the x values that we plug into our function, what is the range of y values that we get back? the important thing to notice here is that 2 to the x, it's an exponential function.

00:56 And exponential functions never are going to have a negative, we're never going to get a negative value back for that exponential function.

01:05 And because of that, this 2 to the x value is always going to be positive.

01:09 And then we're multiplying it by a negative number here.

01:12 And so we're always actually going to get a negative value for this negative 1 times 2 to the x.

01:17 So our range is going to include all of the negative numbers.

01:20 The only thing is it's also going to include positive numbers, and it's going to include positive numbers up to, the number positive 3.

01:28 And the reason for that is because exponential functions, whenever we raise a number to an exponent, we can always guarantee that we're never going to get a value equal to zero back.

01:38 We're equal to exactly zero.

01:40 And it might be very close to zero, but we're never going to get a value equal to exactly zero back.

01:45 So that means that the lowest value we can get here for this negative 1 times 2 to the x is a value almost equal to zero, which means the highest value for our range is going to be three.

01:57 However, we're not including it, since we don't actually have a value for our function where we're going to get negative 1 times 2 to the x equaling 0, so we don't include 3.

02:07 And so this is our domain and range.

02:09 Now we just want to graph our function.

02:13 So i'm just going to plot, we're just going to be doing, let's do five points here...

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