# Video: Pack 1 • Paper 2 • Question 23

Pack 1 • Paper 2 • Question 23

04:40

### Video Transcript

The functions 𝑓 and 𝑔 have the equations 𝑓 of 𝑥 is equal to five minus 𝑥 divided by two and 𝑔 of 𝑥 is equal to three 𝑥 plus two. There are three parts to this question. a) Find the value of 𝑓 of 15. b) Find 𝑔 minus one of 𝑥. And c) Show that 𝑓𝑓 of 𝑥 is equal to five plus 𝑥 divided by four.

In order to find the value of 𝑓 of 15, we need to substitute 15 into the equation 𝑓 of 𝑥. Replacing 𝑥 with 15 gives us five minus 15 divided by two. Five minus 15 is equal to negative 10. So we have negative 10 divided by two. This is equal to negative five. Therefore, 𝑓 of 15 when 𝑓 of 𝑥 is equal to five minus 𝑥 divided by two is equal to negative five.

Part b asked us to find the function 𝑔 minus one of 𝑥. This is the same as saying the inverse of 𝑔 of 𝑥. In order to work out the inverse of any function, we firstly let 𝑦 equal this function. In this case, 𝑦 is equal to three 𝑥 plus two. We now need to rearrange this equation to make 𝑥 the subject. This means to get 𝑥 on its own. Subtracting two from both sides of the equation gives us 𝑦 minus two is equal to three 𝑥. And dividing both sides by three gives us 𝑦 minus two divided by three is equal to 𝑥.

Rearranging 𝑦 equals three 𝑥 plus two to make 𝑥 the subject gives us 𝑥 is equal to 𝑦 minus two divided by three. This means that the inverse or the opposite of the function 𝑔 of 𝑥 is 𝑔 minus one of 𝑥 which is equal to 𝑥 minus two divided by three. We could check this answer using function machines. Three 𝑥 plus two means that we need to multiply our 𝑥-value by three and then add two. The inverse or opposite of this function would be subtracting two and then dividing by three. This means that the inverse function is 𝑥 minus two divided by three. Therefore, our answer was correct.

The third and final part of this question asked us to show that 𝑓𝑓 of 𝑥 is equal to five plus 𝑥 divided by four. The function 𝑓𝑓 of 𝑥 can be found by substituting 𝑓 of 𝑥 — in this case five minus 𝑥 divided by two — everywhere we see an 𝑥 in the function 𝑓 of 𝑥. This gives us five minus five minus 𝑥 divided by two all over two. As five is the same as 10 over two, we can factorize out a half to give us a half multiplied by 10 over two minus five minus 𝑥 divided by two. As we have a common denominator of two inside the bracket, this can be rewritten as a half multiplied by 10 minus five plus 𝑥 all over two.

Note here that inside the bracket we now have plus 𝑥 as we are subtracting negative 𝑥. The two negatives become a positive. Grouping the terms inside the bracket gives us five plus 𝑥 divided by two as 10 minus five is equal to five. Finally, we can multiply the bracket by a half: a half multiplied by a half is a quarter. Therefore, we are left with five plus 𝑥 divided by four. The function 𝑓𝑓 of 𝑥 is equal to five plus 𝑥 divided by four.