In Exercises 19–32, find the (a) domain and (b) range.

Question

𝔂 = cos(x - 3) + 1

Accepted Answer

Welcome back, everyone. What is the domain and range of the function G of X equals sign of x minus 5 + 1? For this problem we want to recall the basic properties of sign. The domain of sine of X. Corresponds to all real numbers. So X belongs to all real numbers. So simply speaking X belongs to the interval from negative infinity up to infinity. We can take any X that we want to evaluate the value of sign, right? So in the context of this problem, we have sin. Of X minus 5 + 1. That's our function G of X, right? Now we have to understand that subtracting phi from X does not change the domain because this is a horizontal shift. Now we're adding one which changes the Y coordinate because this is a vertical shift. So the vertical shift doesn't affect the domain and the horizontal shift also doesn't affect the domain in the context of a sine function, meaning the domain of G of X. Is also a belongs to. The interval from negative infinity up to infinity. If we're subtracting 5, well, we can still choose any x value that we want to valid sign, right? Well then, so now that we have our domain, let's consider the range. Let's recall that sine oscillates between -1 and 1, so we can write an inequality. -1 is less than or equal to sin of x, and sin of x is less than or equal to 1. It shows us that sin of x oscillates between -1 and 1. That's the range of a sine function. And now what we're going to do is simply rewrite it as -1, is less than or equal to sin of x minus 5, which is less than or equal to 1. The horizontal shift doesn't affect the range. And now the only thing that we're missing is one, right? We need to add one. So considering all three sides of this inequality, we're going to add 1 to each side, -1 + 1 gives us 0. Which is less than or equal to. Sign of x minus 5 + 1. Which is less than or equal to 1 + 1, and that's 2. So we have got our range. Our function is bounded by 0 and 2. Meaning our range is Y belongs to the interval from zero inclusive up to 2 inclusive. And those will be our final answers to this problem. Let's label them. And conclude the video. Thank you for watching.

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = cos(x - 3) + 1

Key Concepts

Domain of a Function

Range of a Function

Transformation of Functions

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