# Question Video: Recognizing the Relationship between the Input and Output of a Function Mathematics • 9th Grade

Given the function 𝑓, the meaning of 𝑓(𝑎 − 1) is “the output when the input is 1 less than 𝑎”. Interpret the following. 1) 𝑓(𝑏 + 3). 2) 𝑓(𝑠) − 3. 3) 𝑓(3 − 𝑥). 4) 𝑓(𝑏) − 𝑓(𝑎). 5) 𝑓(3𝑡). 6) 𝑓(𝑎)^𝑏.

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### Video Transcript

Given the function 𝑓, the meaning of 𝑓 of 𝑎 minus one is the output when the input is one less than 𝑎. Interpret the following. 𝑓 of 𝑏 plus three, 𝑓 of 𝑠 minus three, 𝑓 of three minus 𝑥, 𝑓 of 𝑏 minus 𝑓 of 𝑎, 𝑓 of three 𝑡, and 𝑓 of 𝑎 to the power 𝑏.

Before starting this question, it is worth recalling what we mean by a function 𝑓. If we have any function 𝑓 of 𝑥, then 𝑥 is the input and 𝑓 of 𝑥 is the output. A number inside the bracket affects the input, whereas a number outside of the bracket affects the output. This can be seen from the example, as 𝑎 minus one means one less than 𝑎.

Our first function, 𝑓 of 𝑏 plus three, is very similar to the example. Instead of subtracting one from 𝑎, we’re adding three to 𝑏. This means that 𝑓 of 𝑏 plus three calculates the output when the input is three more than 𝑏.

Our second function, 𝑓 of 𝑠 minus three, is slightly different. This time, the three that is being subtracted is outside of the bracket. 𝑓 of 𝑠 will be the output when the input is 𝑠. Therefore, 𝑓 of 𝑠 minus three is three less than the output when the input is 𝑠.

Our third function, 𝑓 of three minus 𝑥, is very similar to the first one and also the example. This time, our function gives us the output when the input is 𝑥 less than three. We are subtracting 𝑥 from three and then working out the output.

Our fourth function has two variables, 𝑏 and 𝑎. We have 𝑓 of 𝑏 minus 𝑓 of 𝑎. This means that we are subtracting the value of 𝑓 of 𝑎 from the value of 𝑓 of 𝑏. This is the difference between them. Therefore, the answer corresponds to the change in output when the input changes from 𝑎 to 𝑏.

The penultimate function is 𝑓 of three 𝑡. We are multiplying our value of 𝑡 by three. This is a similar idea once again to our first function, 𝑓 of 𝑏 plus three, and also our third function, 𝑓 of three minus 𝑥. This time, it corresponds to the output when the input is three times 𝑡.

Our final function involves exponents or powers. We have 𝑓 of 𝑎 to the power of 𝑏. As the 𝑏 is outside of the bracket or parentheses, this is the result of raising the output at input 𝑎 to the 𝑏th power. If the power 𝑏 was inside the bracket, we would be raising the input to the 𝑏th power. Interpreting and often drawing functions of this type is an important part of the topic.