Now that we’ve covered the basics on functions, let’s try out some extra-challenging function questions! Remember that a function is simply a fancy way of describing an equation, but instead of using (x,y) as our input and output, we’re using (x, f(x)), or if not x, another simply variable or symbol.

The function m(n) is defined by . Which of the following is the value of m(m(n))?

When we have functions inside functions, we call these “nested” functions – it’s important to always start from the innermost parentheses and work your way out. The expression m(m(n)) starts with an inntermost “m(n)” that we know equals , so let’s start by substituting that in. Now the question is asking, what is the value of . Just like we normally do for any f(x) function, we will plug whatever in inside the parentheses in for the variable in the function. Here, the testmaker is trying to confuse you by using the same variable “n,” but we’re too smart for the ACT’s tricks! 🙂

According to the order of operations, or PEMDAS, we’d start with what is inside the parentheses. Since we don’t know the value of “n” we cannot simplify  any further. Next, we simplify the exponents. Here’s where knowing your exponent rules really come in handy! When two exponents are separated by a parenthesis, we must always multiply them.

Remember that in number properties, any negative number raised to an odd exponent will stay negative, since it takes a pair of negatives to cancel each other out and become positive. will stay negative until it is multiplied by the -1 in front of the parentheses. Finally, we arrive at our answer:

The correct answer is (E). Let’s try an even harder problem involving functions!

The function values for p(x) vary directly as x for all real numbers. Which of the following best describes the graph of y = p(x) in the standard (x,y) coordinate plane?

(F) A line with a y-intercept of 0.

(G) A line with a y-intercept not at 0.

(H) A line with no y-intercept.

(J) A hyperbola.

(K) Neither a line nor a hyperbola.

The function values for p(x) vary directly as x for all real numbers.The graph of y=p(x) in a standard (x,y) coordinate plane is a line with a y-intercept at 0. Since y = p(x) doesn’t have any coefficients, and p(x) varies directly with x (like p(x) = x, as opposed to something like p(x) = 2x), this line will cross the origin (0,0). The correct answer is (F).

So why were these questions so challenging? Because the testmaker was trying to combine the concept of Functions with other concepts such as PEMDAS, exponent rules, and coordinate geometry. Be on the lookout for these tough “combo” questions – you’ll need to be able to integrate your knowledge of multiple ACT Math concepts to get them right!