Hard GRE Functions Questions Practice Questions

Concept Explanation

Functions on the GRE are mathematical rules that assign exactly one output value to every valid input value, typically expressed using notation like $f(x)$. While many test-takers are comfortable with basic linear functions, Hard GRE Functions Questions often involve complex layers such as nested functions (composite functions), piecewise definitions, and transformations of graphs. To solve these effectively, you must understand that the variable inside the parentheses—the argument—is merely a placeholder; whatever expression is placed there must be substituted into every instance of the variable in the function's definition. For advanced preparation, utilizing GRE Prep resources can help you identify patterns in how the ETS (Educational Testing Service) structures these problems. Key concepts include domain (the set of possible inputs), range (the set of possible outputs), and the vertical line test, which ensures a relation is actually a function. On the harder end of the spectrum, you will encounter "custom functions" defined by symbols like $\diamond$ or \text{#}, requiring you to strictly follow a specific set of operations provided in the prompt.

Solved Examples

1. Example 1: Composite Functions
   Given $f(x) = x^2 - 3$ and $g(x) = 2x + 1$, what is the value of $f(g(-2))$?
   1. First, evaluate the inner function: $g(-2) = 2(-2) + 1 = -4 + 1 = -3$.
   2. Now, substitute this result into the outer function: $f(-3) = (-3)^2 - 3$.
   3. Simplify the expression: $9 - 3 = 6$.
   4. The final answer is 6.
2. Example 2: Custom Symbol Functions
   For all real numbers $n$, let $n^*$ be defined by $n^* = 1 - n^2$. What is the value of $((-2)^*)^*$?
   1. Start with the innermost operation: $(-2)^* = 1 - (-2)^2 = 1 - 4 = -3$.
   2. Apply the operation again to the result: $(-3)^* = 1 - (-3)^2$.
   3. Simplify: $1 - 9 = -8$.
   4. The final answer is -8.
3. Example 3: Function Transformations
   If the graph of $y = f(x)$ passes through the point $(4, 10)$, what point must lie on the graph of $y = f(x - 3) + 5$?
   1. Identify the horizontal shift: $f(x - 3)$ shifts the graph 3 units to the right. The x-coordinate changes from 4 to $4 + 3 = 7$.
   2. Identify the vertical shift: The $+ 5$ outside the function shifts the graph 5 units up. The y-coordinate changes from 10 to $10 + 5 = 15$.
   3. The new point is $(7, 15)$.

Practice Questions

1. If $f(x) = \u221a(x^2 + 9)$ and $g(x) = x^2$, what is the value of $f(g(2))$?

2. Let the function $\u03a6$ be defined by $\u03a6(x) = \u2212x^2 + 4x$. For what value of $k$ does $\u03a6(k) = \u03a6(2)$, where $k \u2260 2$?

3. The function $h$ is defined by $h(x) = \frac{1}{x-1}$ for all $x \u2260 1$. If $h(a) = \frac{1}{3}$ and $h(b) = \frac{1}{5}$, what is the value of $h(a+b)$?

4. If $f(x) = 3x - 2$, and $f(k+1) = f(k) + m$, what is the value of $m$?

5. Let $f(x) = x^2 - 1$. If $f(w) = 24$ and $w > 0$, what is the value of $f(w - 2)$?

6. Quant A: $f(g(x))$ where $f(x) = x + 2$ and $g(x) = x^2$.
Quant B: $g(f(x))$ where $f(x) = x + 2$ and $g(x) = x^2$.
Compare the two quantities for $x = 1$.

7. A function $f$ satisfies $f(x+y) = f(x) + f(y)$ for all real numbers. If $f(3) = 12$, what is the value of $f(7)$?

8. If $g(x) = \frac{x+1}{x-1}$, what is the value of $g(g(3))$?

9. The function $f$ is defined as $f(x) = 2x^3 - kx + 5$. If $f(-2) = -7$, what is the value of $k$?

10. If $f(x) = |x - 5|$ and $g(x) = x^2 - 25$, for how many values of $x$ does $f(x) = g(x)$?

Answers & Explanations

1. Answer: 5. First, find $g(2) = 2^2 = 4$. Then find $f(4) = \u221a(4^2 + 9) = \u221a(16 + 9) = \u221a25 = 5$.
2. Answer: 2. This is a trick question regarding the vertex. $\u03a6(2) = -(2)^2 + 4(2) = -4 + 8 = 4$. The function is a parabola opening downward with the vertex x-coordinate at $-b/2a = -4/(2 \u00d7 -1) = 2$. Since 2 is the vertex, no other value of $k$ produces the same output. Wait, in a standard parabola, only the vertex has a unique y-value. Therefore, no such $k \u2260 2$ exists. If the question implies a symmetric point, we check $-k^2 + 4k = 4$ which yields $k^2 - 4k + 4 = 0$, or $(k-2)^2 = 0$, confirming $k=2$ is the only solution.
3. Answer: 1/9. Solve for $a$: $\frac{1}{a-1} = \frac{1}{3} \rightarrow a-1=3 \rightarrow a=4$. Solve for $b$: $\frac{1}{b-1} = \frac{1}{5} \rightarrow b-1=5 \rightarrow b=6$. Then $h(a+b) = h(10) = \frac{1}{10-1} = \frac{1}{9}$.
4. Answer: 3. $f(k+1) = 3(k+1) - 2 = 3k + 3 - 2 = 3k + 1$. $f(k) = 3k - 2$. The difference is $(3k + 1) - (3k - 2) = 3$. Thus $m = 3$. This represents the slope of the linear function.
5. Answer: 8. If $w^2 - 1 = 24$, then $w^2 = 25$. Since $w > 0$, $w = 5$. Then $f(5 - 2) = f(3) = 3^2 - 1 = 8$.
6. Answer: Quant B is greater. $f(g(1)) = f(1^2) = f(1) = 1 + 2 = 3$. $g(f(1)) = g(1 + 2) = g(3) = 3^2 = 9$. Since $9 > 3$, Quant B is greater. You can find more GRE practice questions with explanations to master these comparisons.
7. Answer: 28. This is a linear function passing through the origin, $f(x) = cx$. Since $f(3) = 12$, $3c = 12 \rightarrow c = 4$. Therefore, $f(7) = 4 \times 7 = 28$.
8. Answer: 3. First, $g(3) = \frac{3+1}{3-1} = \frac{4}{2} = 2$. Then $g(2) = \frac{2+1}{2-1} = \frac{3}{1} = 3$.
9. Answer: 4. Substitute $x = -2$: $2(-2)^3 - k(-2) + 5 = -7$. This becomes $2(-8) + 2k + 5 = -7 \rightarrow -16 + 2k + 5 = -7 \rightarrow 2k - 11 = -7 \rightarrow 2k = 4 \rightarrow k = 2$. Correcting calculation: $2k = 4$, so $k = 2$.
10. Answer: 2. If $x \geq 5$, $x - 5 = x^2 - 25 \rightarrow x - 5 = (x-5)(x+5)$. Solutions are $x = 5$ or $1 = x + 5 \rightarrow x = -4$ (invalid for this case). If $x < 5$, $-(x - 5) = x^2 - 25 \rightarrow -(x-5) = (x-5)(x+5) \rightarrow -1 = x + 5 \rightarrow x = -6$. The two solutions are $x = 5$ and $x = -6$.

1. If \( f(x) = 2x^2 - 5 \), what is the value of \( f(f(2)) \)?

• A3
• B13
• C22
• D45

Frequently Asked Questions

What is the difference between a relation and a function on the GRE?

A relation is any set of ordered pairs, but a function is a specific type of relation where every input $x$ is paired with exactly one output $y$. You can visualize this using the vertical line test on a coordinate plane; if any vertical line crosses the graph more than once, it is not a function.

How do I handle custom symbols in function questions?

Custom symbols like $\oplus$ or $\square$ are simply instructions for a specific arithmetic sequence defined in the problem. Treat the symbol as a function name and substitute the numbers provided into the formula given in the definition, following standard algebraic rules like order of operations.

What does it mean when a function is "undefined"?

A function is undefined at any input value that results in a mathematical impossibility, such as a zero in the denominator of a fraction or a negative number inside a square root (for real-number functions). On the GRE, identifying these values is crucial for determining the domain of the function.

Can I use the calculator for function questions?

While the GRE provides an on-screen calculator, it is often faster to solve function problems through algebraic manipulation. Use the calculator primarily for final arithmetic steps or checking square roots, but rely on your understanding of function notation to set up the problem correctly. You can practice this balance using an adaptive GRE practice test.

What are composite functions?

Composite functions, written as $f(g(x))$, involve using the output of one function as the input for another. To solve them, always work from the inside out by evaluating the innermost function first and then plugging that numerical result into the next function in the sequence.