Question

complete the table of values for the functions $f(x) = 1.5x$ and $g(x) = -(x + 1)^2$. write your answers as whole numbers, decimals, or simplified fractions or mixed numbers.\

$x$	$f(x)$	$g(x)$	\
---	---	---	\
$-2$	$\square$	$\square$	\
$-1$	$\square$	$\square$	\
$0$	$0$	$-1$	\
$1$	$\square$	$\square$	\
$2$	$3$	$-9$	\

based on the values in the table, where does the equation $f(x) = g(x)$ have a solution?\
between $x = -1$ and $x = 0$\
$x = 0$\
between $x = 0$ and $x = 1$\
$x = 1$

Explanation:

Part 1: Completing the table for \( f(x) = 1.5x \) and \( g(x) = -(x + 1)^2 \)

For \( f(x) = 1.5x \):

• When \( x = -2 \):

Step 1: Substitute \( x = -2 \) into \( f(x) \)

\( f(-2) = 1.5 \times (-2) \)

Step 2: Calculate the value

\( 1.5 \times (-2) = -3 \)

• When \( x = -1 \):

Step 1: Substitute \( x = -1 \) into \( f(x) \)

\( f(-1) = 1.5 \times (-1) \)

Step 2: Calculate the value

\( 1.5 \times (-1) = -1.5 \) (or \( -\frac{3}{2} \))

• When \( x = 1 \):

Step 1: Substitute \( x = 1 \) into \( f(x) \)

\( f(1) = 1.5 \times 1 \)

Step 2: Calculate the value

\( 1.5 \times 1 = 1.5 \) (or \( \frac{3}{2} \))

For \( g(x) = -(x + 1)^2 \):

• When \( x = -2 \):

Step 1: Substitute \( x = -2 \) into \( g(x) \)

\( g(-2) = -(-2 + 1)^2 \)

Step 2: Simplify inside the parentheses

\( -(-1)^2 \)

Step 3: Calculate the square and the negative

\( - (1) = -1 \)

• When \( x = -1 \):

Step 1: Substitute \( x = -1 \) into \( g(x) \)

\( g(-1) = -(-1 + 1)^2 \)

Step 2: Simplify inside the parentheses

\( - (0)^2 \)

Step 3: Calculate the square and the negative

\( - 0 = 0 \)

• When \( x = 1 \):

Step 1: Substitute \( x = 1 \) into \( g(x) \)

\( g(1) = -(1 + 1)^2 \)

Step 2: Simplify inside the parentheses

\( - (2)^2 \)

Step 3: Calculate the square and the negative

\( - 4 = -4 \)

Filled Table:

\( x \)	\( f(x) \)	\( g(x) \)
-1	-1.5	0
0	0	-1
1	1.5	-4
2	3	-9

Part 2: Finding where \( f(x) = g(x) \)

To determine where \( f(x) = g(x) \), we analyze the values of \( f(x) \) and \( g(x) \) at the given \( x \)-values:

• At \( x = -1 \): \( f(-1) = -1.5 \), \( g(-1) = 0 \) (so \( f(x) < g(x) \))
• At \( x = 0 \): \( f(0) = 0 \), \( g(0) = -1 \) (so \( f(x) > g(x) \))
• At \( x = 1 \): \( f(1) = 1.5 \), \( g(1) = -4 \) (so \( f(x) > g(x) \))

Since \( f(x) \) goes from less than \( g(x) \) at \( x = -1 \) to greater than \( g(x) \) at \( x = 0 \), by the Intermediate Value Theorem, there must be a solution between \( x = -1 \) and \( x = 0 \).

Final Table Values:

• \( f(-2) = -3 \), \( g(-2) = -1 \)
• \( f(-1) = -1.5 \), \( g(-1) = 0 \)
• \( f(1) = 1.5 \), \( g(1) = -4 \)

Solution to \( f(x) = g(x) \):

The equation \( f(x) = g(x) \) has a solution between \( x = -1 \) and \( x = 0 \).

Answer:

s:

Table Completion:

• \( x = -2 \): \( f(x) = \boldsymbol{-3} \), \( g(x) = \boldsymbol{-1} \)
• \( x = -1 \): \( f(x) = \boldsymbol{-1.5} \) (or \( -\frac{3}{2} \)), \( g(x) = \boldsymbol{0} \)
• \( x = 1 \): \( f(x) = \boldsymbol{1.5} \) (or \( \frac{3}{2} \)), \( g(x) = \boldsymbol{-4} \)

Solution to \( f(x) = g(x) \):

The solution is between \( x = -1 \) and \( x = 0 \), so the answer is: