Functions: complete GMAT guide

A function assigns exactly one output to each allowed input. The notation $f(x)$ means the output of function $f$ when the input is $x$. If $f(x)=2x+3$, then $f(4)=2(4)+3=11$. The parentheses do not mean multiplication. $f(4)$ is the value of the function at 4, not $f$ times 4.

To evaluate a function at an expression, replace every $x$ with that expression. If $f(x)=x^2-1$, then $f(a+b)=(a+b)^2-1$. Keep the entire expression inside parentheses when you substitute.

The domain is the set of allowed inputs. The range is the set of possible outputs. GMAT questions rarely ask for a formal domain, but they do test restrictions:

• A denominator cannot be zero.
• An even root requires a nonnegative expression under the radical.
• A real-world input such as time or a count cannot be negative.

For $f(x)=1/(x-3)$, the domain is all real numbers except 3. For $f(x)=\sqrt{x-2}$, the domain is $x \ge 2$.

A linear function has the form $f(x)=mx+b$. The graph is a line with slope $m$ and $y$-intercept $b$. The slope measures the change in output per unit change in input:

A positive slope means the function increases as $x$ increases. A negative slope means it decreases. A zero slope means the function is constant.

A quadratic function has the form $f(x)=ax^2+bx+c$, with $a \neq 0$. Its graph is a parabola. If $a \gt 0$, the parabola opens upward and has a minimum. If $a \lt 0$, it opens downward and has a maximum. The vertex is at

The vertex gives the input that produces the minimum or maximum output. To find the output, substitute that $x$ into the function.

The function $f(x)=|x|$ returns the distance of $x$ from zero. Its graph is V-shaped, with a minimum at the vertex. For $f(x)=|x-4|+2$, the vertex is at $(4,2)$, and the minimum value is 2. Absolute value is never negative.

Equations with absolute value can have two solutions. $|x-3|=5$ means $x-3=5$ or $x-3=-5$, so $x=8$ or $x=-2$.

A composite function applies one function to the output of another. The notation $f(g(x))$ means evaluate $g$ first, then apply $f$ to the result. Work from the inside out.

Function composition is not generally commutative. $f(g(x))$ and $g(f(x))$ usually produce different results. If a question asks for one of them, do not assume the other is the same.

An inverse function reverses the input-output relationship. If $f(3)=7$, then $f^{-1}(7)=3$. To find an inverse algebraically, replace $f(x)$ with $y$, swap $x$ and $y$, then solve for $y$.

The notation $f^{-1}$ does not mean $1/f$. It means the inverse function, when the inverse exists.

Some questions define a function through a table, a graph, or a custom symbol. Read the definition exactly. If the symbol $\star$ is defined by $a \star b=a^2-b$, then $3 \star 4=9-4=5$. The order of the inputs matters unless the definition is symmetric.

For table questions, locate the input in the first column and read the corresponding output. For graph questions, find the $x$-coordinate on the horizontal axis and read the $y$-coordinate from the curve.

For a function $f(x)$:

• $f(x)+k$ shifts the graph up by $k$.
• $f(x)-k$ shifts it down by $k$.
• $f(x-h)$ shifts it right by $h$.
• $f(x+h)$ shifts it left by $h$.
• $-f(x)$ reflects it across the $x$-axis.
• $f(-x)$ reflects it across the $y$-axis.

Horizontal shifts feel reversed: $f(x-3)$ moves the graph right, not left. This is because the input must be 3 larger to produce the same output.

Some questions define a function with an unknown constant, such as $f(x)=ax^2+2$, and give one value of the function. Use the given value to solve for the constant, then evaluate the function at the requested input. Do not assume the constant is 1 or 0.

For custom symbols, the definition may involve two variables or a condition. Read the order of operations carefully. If $a \star b=2a-b$, then $3 \star 4=6-4=2$, while $4 \star 3=8-3=5$. The operation is not commutative, so the order of the inputs matters.

A graph of a function shows the input on the horizontal axis and the output on the vertical axis. To evaluate the function at a point, find the input on the horizontal axis and read the corresponding output. To solve $f(x)=0$, find where the graph crosses the horizontal axis. To solve $f(x)=k$, draw a horizontal line at $k$ and find the intersections.

A table works the same way. Locate the input in the first column and read the output in the second. If the table does not list the input, the function may be defined by a formula or the question may ask you to infer the pattern. Do not assume a linear pattern unless the table shows a constant difference.

1. Read the definition of the function carefully, including any restrictions.
2. Substitute the input with parentheses around the entire expression.
3. For composites, evaluate the inner function first.
4. For maxima or minima, find the vertex or test the relevant values.
5. For inverse questions, swap the variables and solve.
6. Check the domain and whether the answer makes sense.

• Treating $f(x)$ as multiplication.
• Substituting without parentheses, especially with negative inputs.
• Assuming $f(g(x))=g(f(x))$.
• Confusing $f^{-1}(x)$ with $1/f(x)$.
• Reversing the direction of a horizontal shift.
• Forgetting that an absolute value equation can have two solutions.
• Ignoring a denominator or radical restriction.