Question

complete the table of values for the functions $f(x) = \frac{5}{4}x$ and $g(x) = \frac{1}{4}x^2 + 1$.
write your answers as whole numbers, decimals, or simplified fractions or mixed numbers.

$x$	$f(x)$	$g(x)$
$-1$
$0$
$1$
$2$

based on the values in the table, where does the equation $f(x) = g(x)$ have a solution?
$x = 0$ between $x = 0$ and $x = 1$
$x = 1$ between $x = 1$ and $x = 2$

Explanation:

Part 1: Completing the table for \( f(x) = \frac{5}{4}x \) and \( g(x) = \frac{1}{4}x^2 + 1 \)

For \( f(x) = \frac{5}{4}x \):

Step 1: When \( x = -2 \)

Substitute \( x = -2 \) into \( f(x) \):
\( f(-2) = \frac{5}{4} \times (-2) = -\frac{10}{4} = -\frac{5}{2} = -2.5 \)

Step 2: When \( x = -1 \)

Substitute \( x = -1 \) into \( f(x) \):
\( f(-1) = \frac{5}{4} \times (-1) = -\frac{5}{4} = -1.25 \)

Step 3: When \( x = 0 \)

Substitute \( x = 0 \) into \( f(x) \):
\( f(0) = \frac{5}{4} \times 0 = 0 \)

Step 4: When \( x = 1 \)

Substitute \( x = 1 \) into \( f(x) \):
\( f(1) = \frac{5}{4} \times 1 = \frac{5}{4} = 1.25 \)

Step 5: When \( x = 2 \)

Substitute \( x = 2 \) into \( f(x) \):
\( f(2) = \frac{5}{4} \times 2 = \frac{10}{4} = \frac{5}{2} = 2.5 \)

For \( g(x) = \frac{1}{4}x^2 + 1 \):

Step 1: When \( x = -2 \)

Substitute \( x = -2 \) into \( g(x) \):
\( g(-2) = \frac{1}{4} \times (-2)^2 + 1 = \frac{1}{4} \times 4 + 1 = 1 + 1 = 2 \)

Step 2: When \( x = -1 \)

Substitute \( x = -1 \) into \( g(x) \):
\( g(-1) = \frac{1}{4} \times (-1)^2 + 1 = \frac{1}{4} \times 1 + 1 = \frac{1}{4} + 1 = \frac{5}{4} = 1.25 \)

Step 3: When \( x = 0 \)

Substitute \( x = 0 \) into \( g(x) \):
\( g(0) = \frac{1}{4} \times 0^2 + 1 = 0 + 1 = 1 \)

Step 4: When \( x = 1 \)

Substitute \( x = 1 \) into \( g(x) \):
\( g(1) = \frac{1}{4} \times 1^2 + 1 = \frac{1}{4} + 1 = \frac{5}{4} = 1.25 \)

Step 5: When \( x = 2 \)

Substitute \( x = 2 \) into \( g(x) \):
\( g(2) = \frac{1}{4} \times 2^2 + 1 = \frac{1}{4} \times 4 + 1 = 1 + 1 = 2 \)

Filling the table:

\( x \)	\( f(x) \)	\( g(x) \)
\( -1 \)	\( -1.25 \)	\( 1.25 \)
\( 0 \)	\( 0 \)	\( 1 \)
\( 1 \)	\( 1.25 \)	\( 1.25 \)
\( 2 \)	\( 2.5 \)	\( 2 \)

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

From the table:

• At \( x = 0 \): \( f(0) = 0 \), \( g(0) = 1 \) ( \( f(0) < g(0) \) )
• At \( x = 1 \): \( f(1) = 1.25 \), \( g(1) = 1.25 \) ( \( f(1) = g(1) \) ) Wait, but let's check the values between \( x = 1 \) and \( x = 2 \)? Wait, no, at \( x = 1 \), both are equal? Wait, no, in our table, at \( x = 1 \), \( f(1) = 1.25 \) and \( g(1) = 1.25 \), so \( x = 1 \) is a solution? Wait, but let's re - check the calculation for \( g(1) \):

\( g(1)=\frac{1}{4}(1)^2 + 1=\frac{1}{4}+1=\frac{5}{4}=1.25 \), and \( f(1)=\frac{5}{4}(1)=\frac{5}{4}=1.25 \). So \( f(1) = g(1) \), so \( x = 1 \) is a solution. But wait, maybe there is a miscalculation? Wait, no, the calculation is correct. But let's check the options:

Wait, the options are \( x = 0 \), between \( x = 0 \) and \( x = 1 \), \( x = 1 \), between \( x = 1 \) and \( x = 2 \).

From our table, at \( x = 1 \), \( f(x)=g(x) = 1.25 \). So the solution is \( x = 1 \).

Final Answers

Table Completion:

• For \( x=-2 \), \( f(x)=-2.5 \), \( g(x)=2 \)
• For \( x = - 1 \), \( f(x)=-1.25 \), \( g(x)=1.25 \)
• For \( x = 0 \), \( f(x)=0 \), \( g(x)=1 \)
• For \( x = 1 \), \( f(x)=1.25 \), \( g(x)=1.25 \)
• For \( x = 2 \), \( f(x)=2.5 \), \( g(x)=2 \)

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

The equation \( f(x) = g(x) \) has a solution at \( \boldsymbol{x = 1} \)

Answer:

Part 1: Completing the table for \( f(x) = \frac{5}{4}x \) and \( g(x) = \frac{1}{4}x^2 + 1 \)

For \( f(x) = \frac{5}{4}x \):

Step 1: When \( x = -2 \)

Substitute \( x = -2 \) into \( f(x) \):
\( f(-2) = \frac{5}{4} \times (-2) = -\frac{10}{4} = -\frac{5}{2} = -2.5 \)

Step 2: When \( x = -1 \)

Substitute \( x = -1 \) into \( f(x) \):
\( f(-1) = \frac{5}{4} \times (-1) = -\frac{5}{4} = -1.25 \)

Step 3: When \( x = 0 \)

Substitute \( x = 0 \) into \( f(x) \):
\( f(0) = \frac{5}{4} \times 0 = 0 \)

Step 4: When \( x = 1 \)

Substitute \( x = 1 \) into \( f(x) \):
\( f(1) = \frac{5}{4} \times 1 = \frac{5}{4} = 1.25 \)

Step 5: When \( x = 2 \)

Substitute \( x = 2 \) into \( f(x) \):
\( f(2) = \frac{5}{4} \times 2 = \frac{10}{4} = \frac{5}{2} = 2.5 \)

For \( g(x) = \frac{1}{4}x^2 + 1 \):

Step 1: When \( x = -2 \)

Substitute \( x = -2 \) into \( g(x) \):
\( g(-2) = \frac{1}{4} \times (-2)^2 + 1 = \frac{1}{4} \times 4 + 1 = 1 + 1 = 2 \)

Step 2: When \( x = -1 \)

Substitute \( x = -1 \) into \( g(x) \):
\( g(-1) = \frac{1}{4} \times (-1)^2 + 1 = \frac{1}{4} \times 1 + 1 = \frac{1}{4} + 1 = \frac{5}{4} = 1.25 \)

Step 3: When \( x = 0 \)

Substitute \( x = 0 \) into \( g(x) \):
\( g(0) = \frac{1}{4} \times 0^2 + 1 = 0 + 1 = 1 \)

Step 4: When \( x = 1 \)

Substitute \( x = 1 \) into \( g(x) \):
\( g(1) = \frac{1}{4} \times 1^2 + 1 = \frac{1}{4} + 1 = \frac{5}{4} = 1.25 \)

Step 5: When \( x = 2 \)

Substitute \( x = 2 \) into \( g(x) \):
\( g(2) = \frac{1}{4} \times 2^2 + 1 = \frac{1}{4} \times 4 + 1 = 1 + 1 = 2 \)

Filling the table:

\( x \)	\( f(x) \)	\( g(x) \)
\( -1 \)	\( -1.25 \)	\( 1.25 \)
\( 0 \)	\( 0 \)	\( 1 \)
\( 1 \)	\( 1.25 \)	\( 1.25 \)
\( 2 \)	\( 2.5 \)	\( 2 \)

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

From the table:

• At \( x = 0 \): \( f(0) = 0 \), \( g(0) = 1 \) ( \( f(0) < g(0) \) )
• At \( x = 1 \): \( f(1) = 1.25 \), \( g(1) = 1.25 \) ( \( f(1) = g(1) \) ) Wait, but let's check the values between \( x = 1 \) and \( x = 2 \)? Wait, no, at \( x = 1 \), both are equal? Wait, no, in our table, at \( x = 1 \), \( f(1) = 1.25 \) and \( g(1) = 1.25 \), so \( x = 1 \) is a solution? Wait, but let's re - check the calculation for \( g(1) \):

\( g(1)=\frac{1}{4}(1)^2 + 1=\frac{1}{4}+1=\frac{5}{4}=1.25 \), and \( f(1)=\frac{5}{4}(1)=\frac{5}{4}=1.25 \). So \( f(1) = g(1) \), so \( x = 1 \) is a solution. But wait, maybe there is a miscalculation? Wait, no, the calculation is correct. But let's check the options:

Wait, the options are \( x = 0 \), between \( x = 0 \) and \( x = 1 \), \( x = 1 \), between \( x = 1 \) and \( x = 2 \).

From our table, at \( x = 1 \), \( f(x)=g(x) = 1.25 \). So the solution is \( x = 1 \).

Final Answers

Table Completion:

• For \( x=-2 \), \( f(x)=-2.5 \), \( g(x)=2 \)
• For \( x = - 1 \), \( f(x)=-1.25 \), \( g(x)=1.25 \)
• For \( x = 0 \), \( f(x)=0 \), \( g(x)=1 \)
• For \( x = 1 \), \( f(x)=1.25 \), \( g(x)=1.25 \)
• For \( x = 2 \), \( f(x)=2.5 \), \( g(x)=2 \)

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

The equation \( f(x) = g(x) \) has a solution at \( \boldsymbol{x = 1} \)