Functions
Students struggle with function-based questions on the ACT. This chapter will give you all the information and solved examples you need to get the function questions correct
You put a 2 into the magic box, and a 7 comes out. You put a 3 into the magic box, and a 9 comes out. You put a 4 into the magic box, and an 11 comes out. What is the magic box doing to your number?
There are many possible ways to describe what the magic box is doing to your number. One possibility is as follows: The magic box is doubling your number and adding 3.
Assuming that this is the case (it is possible that the magic box is actually doing something different to your number), this description would yield the following “rule” for this magic box: 2x+3.
This rule can be written in function form as: f(x) = 2x + 3
The function f represents the “rule” that the magic box is using to transform your number. The magic box analogy is a helpful way to understand functions.
Examples of functions include:
- \(f(x)=4x^{2}-11\) The value of the function, f, is dependent on the independent variable,
- \(g(t)=t^{3}+\sqrt{t}-1\) The value of the function, g, is dependent on independent variable, t.
By the way, the expression f(x) is pronounced “f of x”, not “fx.” It does not mean “f times x”! The letter f does not stand for a variable; rather, it stands for the rule that dictates how the input x changes into the output f(x).
Solved Example 1
Some values of the linear function f are shown in the table above. Which of the following defines f?
A. f(x) = 2x + 3
B. f(x) = 3x + 2
C. f(x) = 4x + 1
D. f(x) = 5x
Solution: Oftentimes the best strategy for these types of questions is to plug in answers to make our lives simpler. This way, we don’t actually have to find the equation on our own we can simply test which answer choices match the inputs and outputs we are given in our table.
Let's test the second ordered pair, (3,13) with each answer option. For the correct answer, when we plug the x-value (3) into the equation, we'll end up with the correct y-value (13).
A) f(x) = 2(3) + 3 = 9. This equation is incorrect since 9 doesn't equal 13.
B) f(x) = 3(3) + 2 = 1. This equation is also incorrect.
C) f(x) = 4(3) + 1 = 13. It's a match!
D) f(x) = 5(3) = 15. This equation is also incorrect.
Everything else apart from C gets eliminated. The correct answer is C.
Domains and Range
All x-coordinates are called the DOMAIN, and the y-coordinates are called the RANGE.
For Example: In the following relation: (-5,-3) (-2,0) (1,3) (4,6) (7, 9)
- DOMAIN (all the x-values): {-5, -2, 1, 4, 7}
- RANGE (all the y-values): {-3, 0, 3, 6, 9}
Sometimes, when we are given several ordered pairs, we can connect them by drawing a straight line through all the points.
Example: Graph the relation (-5, -3) (-2, 0) (1, 3) (4, 6) (7, 9) by drawing a line:
For a function, there is only one y-value for each x-value. In other words, a function is a kind of relationship where none of the x-values repeat. In the example above, none of the x-values repeat, so this line represents a FUNCTION.
You can easily determine if a relation is a function by graphing it and doing a VERTICAL LINE TEST. Just draw a vertical line on the graph - if your vertical line ever touches two points of the relation, it's not a function.
Don’t forget that there are two types of variables in equations:
- Independent Variables: can stand alone and aren't affected by other variables.
- Dependent Variables: depends on the independent variables.
With functions, y is the dependent variable - meaning the value of y depends on the value of x.
Example: Is the relation shown in the table a function? Graph it to check your answer.
Is this a function? No, because there are values in the domain that repeat.
Does it pass the vertical line test? No. A vertical line can touch two or more points, so this is NOT a function.
Numerical Substitutions
This is the most basic type of function problem. Input the numerical value (say, 5) in place of the independent variable (x) to determine the value of the function.
Example: If \(f(x)=x^{2}-2\), what is the value of f(5)?
In this problem, you are given a rule for f(x): square x and subtract 2. Then, you are asked to apply this rule to the number 5. Square 5 and subtract 2 from the result: \(f(5)=(5)^{2}-2=25-2=23\)
Solved Example 2
Let the function f be defined by f(x) = 5x - 2a, where a is a constant. If f(10) + f(5) = 55, what is the value of a?
A. -5
B. 0
C. 5
D. 10
Solution: As you can see here, we are given our equation as well as two inputs and their combined output.
f(x) = 5x − 2a
Therefore, f(5) = 5(5) − 2a and f(10) = 5(10) − 2a
f(5) = 25 − 2a and f(10) = 50 − 2a
Now, let us set the sum of our two outputs equal to 55 (as was stipulated in the question).
50 − 2a + 25 − 2a = 55
75 − 4a = 55
- 4a = −20
a = 5
The correct answer is C, a = 5.
Solved Example 3
Which of the following could be the equation of the graph?
A. \(Y=x^{2}+2\)
B. \(Y=(x+2)^{2}\)
C. \(Y=x^{2}-2\)
D. \(Y=(x-2)^{2}\)
Solution: We can see in this graph that the y-intercept is +2. This means that, when we substitute X = 0 in the equation, we get Y = 2.
A. \(Y=0^{2}+2=2\) Keep this option
B. \(Y=(x+2)^{2}=(0+2)^{2}=4\) Eliminate this option
C. \(Y=x^{2}-2=0^{2}-2=-2\) Eliminate this option
D. \(Y=2(x+1)^{2}=2(0+1)^{2}=2\) Keep this option
Only options A and D remain. The easiest way is to plug in numbers from the graph and see what fits for which of the two equations. For example, point (1, 3) is on the graph. Therefore, when we substitute x = 1 in the answer choices, we should get an output of 3.
A. \(Y=1^{2}+2=3\) This is the answer
B. \(Y=2(1+1)^{2}=16\) Eliminate
The correct answer is A.
Algebraic Substitutions
This type of problem is slightly more complicated. Instead of finding the output value for a numerical input, you must find the output when the input is an algebraic expression
Example: If \(f(2)=Y^{2}-4Y\) what is the value of f(w+6)?
Input the variable expression (w+6) in place of the independent variable (z) to determine the value of the function: \(f(w+6)=(w+6)^{2}-4(w+6)\)
The rest is algebraic simplification: \(f(w+6)=(w+6)^{2}-4w-24\)
\(=w^{2}+12w+36-4w-24\)
\(w^{2}+8w+12\)
Compound Functions
Imagine putting a number into one magic box, and then putting the output directly into another magic box. This is the situation you have with COMPOUND FUNCTIONS.
Example: If \(f(x)=x^{3}+g(x)\) and g(x) = 4x - 3, what is f(g(3))?
The expression f(g(3)), pronounced “f of g of 3”, looks ugly, but the key to solving compound function problems is to WORK FROM THE INSIDE OUT. In this case, start with g(3).
g(3) = 4(3) - 3 = 12 - 3 = 9
Use the result from the inner function g as the new input variable for the outer function f: \(f(g(3))=f(9)=(9)^{3}+\sqrt{9}=729+3=732\)
The final result is 732.
Note that changing the order of the compound functions changes the answer.
Example If \(f(x)=x^{3}+g(x)\: and\: g(x)=4x-3,what\: is\: g(f(3))\)
Again, work from the inside out. This time, start with f(3) [which is now the inner function]: \(f(x)=3^{3}+\sqrt{3}=27+\sqrt{3}\)
Use the result from the inner function as the new input variable for the outer function g:
g(f(3)) = g(27 + √3) = 4(27 + √3) - 3
= 108 + 4√3 - 3
= 105 + 4√3
Solved Example 4
What is f(g(x−2)) when \(f(x)=x^{2}-6\) and g(x) = 3x + 4?
A. 3x − 2
B. \(3x^{2}+12x-6\)
C. \(9x^{2}+24x+10\)
D. \(9x^{2}+12x-2\)
Solution: Because g(x) is nested the deepest, we must find its output before we can find f(g(x−2)).
For g(x-2), replace the x in the g(x) function with x-2:
g(x) = 3x + 4
g(x−2) = 3(x−2) + 4
g(x−2) = 3x − 2
So our output of g(x−2) is 3x−2. Again, this is an equation and not an integer, but it still works as an output.
\(f(x)=x^{2}-6\)
\(f(g(x-2))=f(3x-2)=(3x-2)^{2}-6\)
\(=9x^{2}-12x+4-6\)
\(=9x^{2}-12x-2\)
The correct answer is D.
Function Graphs
A function can be visualized by graphing it in the coordinate plane.
Example: What is the graph of the function \(f(x)=-2x^{2}+1?\)
Create a table by evaluating the function for several input values:
Then, plot points to set the shape of the graph:
It may be sufficient to calculate only three or four values as long as you have found the pattern and can accurately represent the graph.
Solved Example 5
The complete graph of function f is shown in the xy-plane. For what value of x is the value of f(x) at its minimum?
A. -5
B. -3
C. -2
D. 3
Solution: The minimum value of a function corresponds to the y-coordinate of the point on the graph where it's lowest on the y-axis.
Looking at the graph, we can see the function's lowest point on the y-axis occurs at (-3,-2). Since we're looking for the value of x when the function is at it's minimum, we need the x-coordinate, which is -3.
The correct answer is B, -3.
