The functions f and g are defined for the integers -2 \leq x \leq 3 by the table below. | x | f(x) | g(x) | | :--- | :--- | :--- | | -2 | 3 | 1 | | -1 | 2 | 3 | | 0 | -2 | -1 | | 1 | 0 | 2 | | 2 | -1 | 0 | | 3 | 1 | 2 | What is the value of x for which g(f(x)) = -1?

The functions f and g are defined for the integers -2 \leq x \leq 3 by the table below. | … — ACT

The functions $f$ and $g$ are defined for the integers $-2 \leq x \leq 3$ by the table below.

What is the value of $x$ for which $g(f(x)) = -1$ ?

The value of

x

1

To solve

g(f(x)) = -1

, we work from the outside in. First, we find the input to the outer function

g

that results in an output of

-1

. Looking at the table,

g(0) = -1

, which means the inner expression

f(x)

must equal

0

. Next, we find the input to

f

that results in an output of

0

. Looking at the table,

f(1) = 0

. Therefore, the value of

x

that satisfies the equation is

1

Identify the outer function in the composition

g(f(x)) = -1

The outer function is

g

, so we must first find an input

u

such that

g(u) = -1

This allows us to break down the composition into two simpler function evaluation steps.

Look at the column for

g(x)

in the table to find where the output is

-1

The table shows

g(0) = -1

, meaning the input to

g

must be

0

. Therefore,

f(x) = 0

This establishes the required output value for the inner function

f(x)

Look at the column for

f(x)

in the table to find where the output is

0

The table shows

f(1) = 0

, meaning the input

x

must be

1

This provides the final value of the variable

x

that satisfies the original equation.

Evaluating composite functions using a table of values

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