Secondary 4 Function Properties and Inverses Worksheet

All 6 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Algebra — Relations and Functions

   Five rules are listed below.

   1. Name the function family of each one (linear, quadratic, square root, exponential, logarithmic, step), or write “not a function” where appropriate.
   2. Exactly one of the five is a relation that is not a function of $x$. Identify it and justify your choice with one specific value of $x$.

   $$y^2=x+5,f(x)=3(2{)}^x,g(x)=\lfloor \frac{1}{2}x\rfloor ,h(x)=-2\sqrt{x+1},k(x)={\log }_{5}x$$

2. Q2Periodic Functions

   A shuttle boat runs a closed loop, so its distance $d$ (in km) from the terminal is a periodic function of the time $t$ (in minutes) with period $50$ minutes. On one cycle, $d(0)=0$, $d(15)=6$, $d(25)=8$ and $d(40)=3$.

   1. Find $d(115)$ and $d(90)$.
   2. If the smallest distance reached is $0\text{ km}$ and the greatest is $8\text{ km}$, give the amplitude of the function.
3. Q3The Inverse of a Function

   A weather station records the function $f=\{(-2,5),(0,1),(3,-4),(6,1)\}$.

   1. Write the inverse relation $f^{-1}$ as a set of ordered pairs.
   2. Is $f^{-1}$ a function? Justify.
4. Q4The Piecewise Function

   A function $f$ is defined on $[-5,5]$ by $$f(x)=\{\begin{array}{cc}-2x-4\hfill & \text{if }-5\le x<-1,\hfill \\ x^2-2\hfill & \text{if }-1\le x<2,\hfill \\ 2\hfill & \text{if }2\le x\le 5.\hfill \end{array}$$

   1. Compute $f(-4)$, $f(-1)$, $f(0)$ and $f(3)$.
   2. Find all the zeroes of $f$.
   3. Sketch the graph of $f$ on the grid below.
   4. Is $f$ continuous at $x=-1$? At $x=2$? Justify.

   A blank Cartesian grid for this question is on the printable PDF.

5. Q5The Properties of Functions

   The graph of a function $g$ is made of three segments: from $(-6,4)$ to $(-2,-4)$, then from $(-2,-4)$ to $(1,-4)$, then from $(1,-4)$ to $(5,6)$. Give the domain, the range, the zeroes, the initial value, the extrema, the intervals of variation and the sign of $g$.

6. Q6The Role of Parameters a, b, h, and k of a Function in Standard Form

   The function $g(x)=-3|2(x+4)|-1$ is obtained from the base function $f(x)=|x|$.

   1. Give the values of $a$, $b$, $h$ and $k$.
   2. Describe, in order, the transformations applied to $f$.
   3. Give the vertex and the range of $g$.

The 6 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Naming the family: look at where the variable sits

Q1 lists several rules and asks for the family of each. The whole classification is decided by one question — where does the variable appear? — and it takes about three seconds per rule once you ask it deliberately.

Q1 also asks a second, separate question: which of the rules is a relation and not a function. A rule written with y squared solves to y = ± something, so one x is paired with two y-values — that is the definition being broken, and the justification the question wants is one specific value of x with its two partners written out.

The property list, in a fixed order

Q5 hands you a graph made of segments and asks for everything. Answering in a fixed order every time is what stops you from omitting one — and an omission here is a straight loss, because each property is marked separately.

Getting exact values off a segment graph

When a zero falls between two grid lines, "about 2.5" is not an answer. Find the rate of change of that segment from its two endpoints, write the segment as y = y₁ + a(x − x₁), set it equal to zero and solve. That is three lines and gives an exact number. Q5 is built with at least one zero that cannot be read off the picture, so the method is not optional.

Periodic functions

A function is periodic with period p when f(t + p) = f(t) for every t. To evaluate it far outside the cycle you were given data for, subtract whole periods until you land back inside it — practically, divide by the period and keep the remainder, then look the remainder up in your data. Q2 asks for two such values.

Piecewise functions: the interval is the instruction

Q4 gives a rule in three branches. Every part of the question is answered by first deciding which branch an x belongs to, and the inequality signs written next to each branch are what decide it — including whether a boundary value belongs to the branch on its left or the one on its right.

• Evaluating. Locate x in the list of intervals, then use only that branch. At a boundary, read the inequality signs carefully: one branch claims it and the other does not.
• Zeroes. Solve each branch's formula separately, then keep only the roots that actually lie inside that branch's interval. A root of a formula that falls outside the stretch where the formula applies is not a zero of the function. Discarding it, with a reason, is the point of the question.
• Continuity at a join. Compare the value the branch on the left is heading towards with the value the function actually takes there. Equal → the pieces meet and the function is continuous. Different → there is a jump, and you say so and name the two values.
• Graphing. Filled dot where a branch includes its endpoint, open dot where it does not — but if the neighbouring branch supplies that value, the point is filled and there is no hole. Both situations occur in Q4, which is why the sketch is worth more than it looks.

Inverses: swap the pairs, then ask about injectivity

Q3 gives a function as a set of ordered pairs, which strips the idea down to its mechanics. The inverse relation is obtained by swapping the two coordinates of every pair — and that is all the inverse ever is, whether the function arrives as a set, a graph or a rule. With a rule you swap x and y and then re-isolate; with a graph you reflect in the line y = x.

Whether the result is a function is decided entirely by f, not by f⁻¹:

So the diagnosis runs backwards from where students look. Find two different inputs of f with the same image and you have proved the inverse is not a function, in one line, by naming them. Q3 is constructed so that such a pair exists and is easy to spot if you look for it.

Two consequences worth writing down every time: the domain of the inverse is the range of f, and the range of the inverse is the domain of f. Those two sentences answer a surprising number of exam questions on their own.

The parameters a, b, h and k

Q6 takes a base function and dresses it in all four parameters at once. Each one does exactly one thing:

• a — vertical scaling by |a|, plus a reflection in the x-axis if a is negative. It also flips a minimum into a maximum, which is what decides the range.
• b — horizontal scaling by 1⁄|b|, plus a reflection in the y-axis if b is negative. Note the inversion: because b multiplies the input, a b bigger than 1 shrinks the graph horizontally. This is the parameter that is described backwards most often.
• h — horizontal translation, h units right when h is positive.
• k — vertical translation, k units up when k is positive.

When the question asks you to describe the transformations in order, put the scalings and reflections first and the translations last. Applied in the other order the numbers no longer match.

Preview all 3 pages

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Getting the most out of it

Answer the property list in the same order every time

Domain, range, zeroes, initial value, extrema, variation, sign. Write the seven headings down the page before you look at the graph, then fill them in. The habit is what stops the two losses that happen on this question — omitting one, and giving the same interval for sign and for variation.

Test every join point twice

On a piecewise rule, evaluate both neighbouring branches at each boundary value. If the two numbers agree the graph is continuous there; if they disagree you have found the jump and you also know which branch owns the point. It answers the evaluation, the sketch and the continuity parts at once.

Write one function three ways

Take any rule on this sheet and produce a table of values, a graph and the rule itself, then read the same property off all three. Moving between representations is the skill this unit is really assessing, and it is not built by staying in one of them.

Common questions