Video Transcript
Consider the function 𝑓 of 𝑥 equals four cos seven 𝑥 plus 𝜋 plus five. What is the domain of 𝑓 of 𝑥? What is the range of 𝑓 of 𝑥?
We will begin by finding the domain of 𝑓 of 𝑥, recalling that the domain of a function is the set of all possible input values. We know that the domain of the cosine function is the set of all real numbers, which we can write as the open interval from negative ∞ to positive ∞. This tells us that there is no restriction for the input values to the cosine function. In the function 𝑓 of 𝑥, the expression seven 𝑥 plus 𝜋 is inside the cosine function. Since this function, seven 𝑥 plus 𝜋, is well defined for any real number, the domain of 𝑓 of 𝑥 will also be the set of all real numbers. In other words, there are no restrictions to the values on which the function 𝑓 can act.
Let’s now consider the range of 𝑓 of 𝑥. We recall that the range of a function is the set of all possible values of the function itself. Or we may think of this as the set of all possible output values given the function’s domain. So, we need to determine the set of all possible values for the expression four cos of seven 𝑥 plus 𝜋 plus five for any real number 𝑥. We know that the range of seven 𝑥 plus 𝜋 is all real numbers, so this expression can take any real value. Let’s denote seven 𝑥 plus 𝜋 by 𝜃 then. And we now need to find the set of possible values for the expression four cos 𝜃 plus five for any real number 𝜃.
We should recall that the range of the cosine function is the closed interval from negative one to positive one. Or, in other words, the cos of 𝜃 is greater than or equal to negative one and less than or equal to positive one. Multiplying by four, we have that four times the cos of 𝜃 is greater than or equal to negative four and less than or equal to four. Then, adding five to each part of the inequality, we have that four cos 𝜃 plus five is greater than or equal to one and less than or equal to nine. The range of the function four cos seven 𝑥 plus 𝜋 plus five is therefore the closed interval from one to nine.
We can also see this by considering the function transformations that have been applied to the cosine function to give four multiplied by cos 𝜃 plus five. Multiplying by four corresponds to a vertical stretch with a scale factor of four. So, this has the effect of increasing the range from the closed interval from negative one to one to the closed interval from negative four to four. Adding five to the function corresponds to a vertical shift by five units upwards. So, this is the effect of adding five to each endpoint of the range.
For the function 𝑓 of 𝑥 equals four cos seven 𝑥 plus 𝜋 plus five then, we found that the domain of 𝑓 of 𝑥 is the open interval from negative ∞ to positive ∞ or the set of all real numbers. And the range of 𝑓 of 𝑥 is the closed interval from one to nine.
