GMAT Math : Functions/Series

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Example Questions

Example Question #61 : Dsq: Understanding Functions

Given a function $\displaystyle f$, it is known that:

$\displaystyle f(0) = 11$

$\displaystyle f(1) = 7$

$\displaystyle f(2) = 10$

$\displaystyle f(3) = 6$

$\displaystyle f(4) = 9$

$\displaystyle f(5) = 8$

Given a function $\displaystyle g$, evaluate $\displaystyle (f g) (0)$.

Statement 1: $\displaystyle g$ is an odd function.

Statement 2: $\displaystyle g (c) = 0$ for every positive integer $\displaystyle c$.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

$(f g) (0) = f(0) \cdot g (0) = 11 \cdot g(0)$ $\displaystyle (f g) (0) = f(0) \cdot g (0) = 11 \cdot g(0)$, so to answer the question, it is necessary and sufficient to evaluate g(0) $\displaystyle g(0)$.

Assume Statement 1 alone. By defintion of an odd function, from Statement 2, for every $\displaystyle a$ in the domain of $\displaystyle g$, $\displaystyle g(-a) = -g(a)$. In specific, setting a = 0 $\displaystyle a = 0$,

$\displaystyle g(0) = -g(0)$.

The only number whose opposite is itself is 0, so

$\displaystyle g(0) = 0$,

and it follows that

$(f g) (0) = 11 \cdot 0 = 0$ $\displaystyle (f g) (0) = 11 \cdot 0 = 0$.

Statement 2 only gives the values of $\displaystyle g$ for positive integers; this information is irrelevant.

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Example Question #62 : Dsq: Understanding Functions

Let $\displaystyle f$ be a function with the set of all real numbers as its domain, and let the function have an inverse $\displaystyle f^{-1}$.

What is the $\displaystyle y$-intercept of the graph of $f^{-1}$ $\displaystyle f^{-1}$?

Statement 1: $\displaystyle f(0) = 6$.

Statement 2: The graphs of $\displaystyle f$ and $f^{-1}$ $\displaystyle f^{-1}$ intersect only at the point (4, 4) $\displaystyle (4, 4)$.

Possible Answers:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation:

The $\displaystyle y$-intercept of the graph of $f^{-1}$ $\displaystyle f^{-1}$ is the point (0, b) $\displaystyle (0, b)$ at which the graph intersects the $\displaystyle y$-axis. At that point, $f^{-1}(0) = b$ $\displaystyle f^{-1}(0) = b$, or, equivalently, $\displaystyle f(b) = 0$.

Therefore, we need to find the value $\displaystyle b$ for which $\displaystyle f(b) = 0$.

Between the two statements, we only know that $\displaystyle f(0) = 6$ and $\displaystyle f(4) = 4$. The value of $\displaystyle b$ for which $\displaystyle f(b) = 0$ cannot be determined.

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Example Question #63 : Dsq: Understanding Functions

True or false: $\left \{ a_{n} \right \}$ $\displaystyle \left \{ a_{n} \right \}$, $\displaystyle n = 1, 2, 3,...$ is an arithmetic sequence.

Statement 1: $a_{1}+ a_{4} \ne a_{2} + a_{3}$ $\displaystyle a_{1}+ a_{4} \ne a_{2} + a_{3}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$ $\displaystyle a_{6}- a_{5} = a_{8} - a_{7}$

Possible Answers:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

Assume Statement 1 alone.

$a_{1}+ a_{4} \ne a_{2} + a_{3}$ $\displaystyle a_{1}+ a_{4} \ne a_{2} + a_{3}$

$a_{1}+ a_{4} - a_{1} - a_{3} \ne a_{2} + a_{3} - a_{1} - a_{3}$ $\displaystyle a_{1}+ a_{4} - a_{1} - a_{3} \ne a_{2} + a_{3} - a_{1} - a_{3}$

$a_{4} - a_{3} \ne a_{2} - a_{1}$ $\displaystyle a_{4} - a_{3} \ne a_{2} - a_{1}$,

meaning that at least two such differences are unequal, and proving the sequence is not arithmetic.

Statement 2 alone only proves that two such differences are equal, but says nothing about any of the other (infinitely many) such differences. Therefore, it leaves the question unresolved.

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Example Question #64 : Dsq: Understanding Functions

Define $\displaystyle f(x)=4x+A$ and $\displaystyle g(x) =Bx+5$.

Is it true that $f = g^{-1}$ $\displaystyle f = g^{-1}$ ?

Statement 1: A = -10 $\displaystyle A = -10$

Statement 2: $B=\frac{1}{2}$ $\displaystyle B=\frac{1}{2}$

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

For $f = g^{-1}$ $\displaystyle f = g^{-1}$ to be each other's inverse, it must be true that

$(f\circ g)(x) =x$ $\displaystyle (f\circ g)(x) =x$ and $(g\circ f)(x) =x$ $\displaystyle (g\circ f)(x) =x$

We can look at the first condition.

$(f\circ g)(x) =x$ $\displaystyle (f\circ g)(x) =x$

$(f\circ g)(x) =f(g(x)) = f(Bx+5)=4(Bx+5)+A=4Bx+20+A= x$ $\displaystyle (f\circ g)(x) =f(g(x)) = f(Bx+5)=4(Bx+5)+A=4Bx+20+A= x$

For this to be true, it must hold that:

4B=1 $\displaystyle 4B=1$

$B=\frac{1}{4}$ $\displaystyle B=\frac{1}{4}$

and

20+A=0 $\displaystyle 20+A=0$

A = -20 $\displaystyle A = -20$

Since both statements violate these conditions, it is impossible for $f = g^{-1}$ $\displaystyle f = g^{-1}$, even if you are only given one of them.

The answer is that either statement alone is sufficient to answer the question.

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GMAT Math : Functions/Series

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #61 : Dsq: Understanding Functions

Example Question #62 : Dsq: Understanding Functions

Example Question #63 : Dsq: Understanding Functions

Example Question #64 : Dsq: Understanding Functions

All GMAT Math Resources

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