Which of the Following Is True About the Function Below: A Complete Guide
You've seen it before. You're staring at a math problem, and there's a function sitting there — maybe it's a graph, maybe it's an equation, maybe it's some hybrid — and the question asks: which of the following is true about the function below?
And suddenly you're scrambling through options (A) through (E), trying to remember what you actually know about functions and hoping something clicks.
Here's the thing — these questions aren't random. There's a logic to them. Practically speaking, once you understand what test-makers are actually checking, these problems become a lot less intimidating. Let me walk you through how to approach them, what properties matter, and how to avoid the mistakes that cost people points The details matter here..
What Are We Actually Looking For?
When a question asks "which of the following is true about the function below," it's asking you to evaluate specific properties of a function. The function might be presented as:
- An equation (like f(x) = x² - 4)
- A graph
- A table of values
- A combination (an equation with a graph provided)
The core skills being tested are your ability to read a function's behavior and check it against mathematical statements. Most of these questions involve checking one or more of these properties:
- Domain — all the x-values the function accepts
- Range — all the possible output values
- Continuity — whether the function has any breaks, holes, or jumps
- Intercepts — where it crosses the x-axis (zeros) or y-axis
- Symmetry — whether it's even, odd, or neither
- Behavior at infinity — what happens as x gets very large or very small
- Differentiability — whether a derivative exists at certain points
Why Does This Matter?
Because these properties tell you everything about how a function behaves. In real terms, in real math — calculus, statistics, modeling — you need to understand a function's limits and capabilities before you use it. Also, knowing whether a function is continuous tells you whether you can apply the Intermediate Value Theorem. Knowing the domain tells you where your calculations are valid Simple, but easy to overlook. Turns out it matters..
In a testing context, these questions also reveal whether you understand the definitions behind the concepts, not just the procedures.
How to Approach Any "Which Is True" Function Question
Here's the step-by-step process that works, regardless of what the function looks like or what the answer choices are Easy to understand, harder to ignore..
Step 1: Identify the Form
Is the function given as an equation, a graph, or both? This determines what information is immediately available to you Worth keeping that in mind..
- Equation: You can algebraically determine domain, intercepts, symmetry, and behavior. You can also rewrite it to reveal properties (like completing the square to find the vertex of a quadratic).
- Graph: You can visually read intercepts, continuity, general shape, and behavior at infinity. Domain and range are often directly observable.
- Both: Use both. The graph confirms what the algebra suggests. Discrepancies between the two usually mean you're misinterpreting something.
Step 2: Check Each Property Systematically
Go through the common properties in order. For each one, either confirm or rule out what the answer choices are claiming.
- Domain: Start with any denominators (can't be zero), square roots or even roots (inside must be non-negative), logarithms (must be positive), and any other restricted operations.
- Range: Find the minimum or maximum values. For basic functions, you can often determine this from the domain restrictions and the function's shape.
- Intercepts: Set x = 0 to find the y-intercept. Set the function equal to 0 and solve for x to find x-intercepts (zeros).
- Symmetry: Check f(-x). If f(-x) = f(x), it's even (symmetric about the y-axis). If f(-x) = -f(x), it's odd (symmetric about the origin). Otherwise, neither.
- Continuity: For graphs, look for holes, jumps, or vertical asymptotes. For equations, check if any domain restrictions create breaks.
Step 3: Eliminate Answer Choices
This is where most people either save time or waste it. Once you've determined what the function actually does, go through each option and cross out the ones that are clearly false Not complicated — just consistent..
The key here: you only need one true statement. Sometimes multiple options seem plausible, but only one is actually correct. That's fine — your job is to find the one that holds up under scrutiny.
Common Mistakes That Trip People Up
Let me tell you about the errors I see most often with these problems. If you can avoid these, you're already ahead.
Confusing Domain and Range
Students constantly mix these up. But the domain is the input (x-values); the range is the output (y-values). Here's the thing — a function can have a restricted domain but still produce a full range of outputs — or vice versa. Don't assume they're the same.
You'll probably want to bookmark this section Small thing, real impact..
Forgetting About Hidden Restrictions
You look at f(x) = √(x + 5) and think "domain is all real numbers." Wrong. Plus, the expression inside the square root must be non-negative, so x + 5 ≥ 0, which means x ≥ -5. Which means these hidden restrictions come from denominators, square roots, logarithms, and rational exponents. Always check And that's really what it comes down to..
Misreading Graphs
Graphs can be deceiving. A curve might look like it hits a certain value, but if there's an open circle (indicating the point is not included), that changes everything. Pay attention to asymptotes, endpoints, and whether points are filled or open Easy to understand, harder to ignore..
Assuming Symmetry Without Checking
Students often see a parabola and assume it's even. So most parabolas are even, but not all. If the vertex is shifted — like f(x) = (x - 3)² — it's still even because f(-x) = f(x). But if there's a linear term added, like f(x) = (x - 3)² + x, that's no longer even. Actually do the f(-x) check when symmetry matters.
Not Checking All Answer Choices
Sometimes (A) looks obviously true, so you pick it and move on. Think about it: always verify at least enough to be confident. But (A) might have a subtle flaw. The right answer isn't always the first one that seems right.
Practical Tips That Actually Help
Here's what I'd tell a student sitting down to practice these problems:
1. Create a quick property checklist. Write down the five or six things you always check (domain, range, intercepts, continuity, symmetry, behavior). Run through it every time. Build the habit.
2. For graph problems, trace with your finger. It sounds simple, but physically following the curve helps you catch intercepts, turning points, and discontinuities you might otherwise miss And that's really what it comes down to..
3. For equation problems, rewrite in helpful forms. Factored form reveals zeros. Vertex form reveals the minimum/maximum. Standard form reveals the y-intercept. If you're stuck, try rewriting.
4. Use the answer choices to guide you. If three of the five options are about the domain, you probably need to focus on domain. If one mentions continuity, that's likely the key property. The options often signal what's important Small thing, real impact..
5. When in doubt, test points. If you're unsure whether a statement is true, pick a value that should reveal the truth. To give you an idea, if an option claims f(2) = 5, plug in x = 2 and check. Concrete examples beat abstract reasoning when you're uncertain Which is the point..
FAQ
How do I quickly find the domain of a function?
Start by identifying operations with restrictions: denominators (can't be zero), even roots (inside must be non-negative), logarithms (argument must be positive). Solve each inequality and take the intersection of all valid x-values.
What's the difference between a function being continuous and being differentiable?
Continuity means there are no breaks, holes, or jumps — the graph is one connected piece. Still, differentiability means the derivative exists at that point, which requires continuity and a smooth tangent (no sharp corners or vertical tangents). Every differentiable function is continuous, but not every continuous function is differentiable Most people skip this — try not to..
How do I check if a function is even or odd?
Calculate f(-x) and compare it to f(x). If f(-x) = f(x), it's even (symmetric about the y-axis). Also, if f(-x) = -f(x), it's odd (symmetric about the origin). Anything else means it's neither.
What if the function is given as a graph and I need to find the equation?
Look for key features: intercepts, vertex (for parabolas), slope (for lines), asymptotes (for rational or exponential functions). Sometimes you can match to a standard form. Other times, you'll need to use points from the graph to solve for coefficients.
Can a function have more than one y-intercept?
No. A function assigns exactly one output to each input. Since x = 0 is a single input, it can only produce one y-value. Functions can have multiple x-intercepts (zeros), but only one y-intercept.
The Bottom Line
These "which of the following is true" questions are really just checking whether you understand the fundamental properties of functions — and whether you can apply that understanding to a specific case. The good news: the properties don't change. Domain, range, intercepts, continuity, symmetry — those are always the building blocks.
Once you know what to look for, you can work through any function, any graph, any equation. The key is being systematic, checking each property against what the answer choices claim, and eliminating anything that doesn't hold up.
Practice with a few problems, and it'll start feeling automatic. You've got this.
