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QUESTION IMAGE

complete the table of values for the functions $f(x) = |x - 3| - 4$ and…

Question

complete the table of values for the functions $f(x) = |x - 3| - 4$ and $g(x) = 3^x - 5$.
write your answers as whole numbers, decimals, or simplified fractions or mixed numbers.

$x$$f(x)$$g(x)$
1
2
3
4

based on the values in the table, where does the equation $f(x) = g(x)$ have a solution?
$x = 1$ between $x = 1$ and $x = 2$
$x = 2$ between $x = 2$ and $x = 3$

Explanation:

Response
Part 1: Completing the table for \( f(x) = |x - 3| - 4 \) and \( g(x) = 3^x - 5 \)
For \( f(x) = |x - 3| - 4 \):
  • When \( x = 0 \):

Step1: Substitute \( x = 0 \) into \( f(x) \)

\( f(0) = |0 - 3| - 4 = |-3| - 4 = 3 - 4 = -1 \)

  • When \( x = 1 \):

Step1: Substitute \( x = 1 \) into \( f(x) \)

\( f(1) = |1 - 3| - 4 = |-2| - 4 = 2 - 4 = -2 \)

  • When \( x = 2 \):

Step1: Substitute \( x = 2 \) into \( f(x) \)

\( f(2) = |2 - 3| - 4 = |-1| - 4 = 1 - 4 = -3 \)

  • When \( x = 3 \):

Step1: Substitute \( x = 3 \) into \( f(x) \)

\( f(3) = |3 - 3| - 4 = |0| - 4 = 0 - 4 = -4 \)

  • When \( x = 4 \):

Step1: Substitute \( x = 4 \) into \( f(x) \)

\( f(4) = |4 - 3| - 4 = |1| - 4 = 1 - 4 = -3 \)

For \( g(x) = 3^x - 5 \):
  • When \( x = 0 \):

Step1: Substitute \( x = 0 \) into \( g(x) \)

\( g(0) = 3^0 - 5 = 1 - 5 = -4 \)

  • When \( x = 1 \):

Step1: Substitute \( x = 1 \) into \( g(x) \)

\( g(1) = 3^1 - 5 = 3 - 5 = -2 \)

  • When \( x = 2 \):

Step1: Substitute \( x = 2 \) into \( g(x) \)

\( g(2) = 3^2 - 5 = 9 - 5 = 4 \)

  • When \( x = 3 \):

Step1: Substitute \( x = 3 \) into \( g(x) \)

\( g(3) = 3^3 - 5 = 27 - 5 = 22 \)

  • When \( x = 4 \):

Step1: Substitute \( x = 4 \) into \( g(x) \)

\( g(4) = 3^4 - 5 = 81 - 5 = 76 \)

Filling the table:
\( x \)\( f(x) \)\( g(x) \)
1-2-2Wait, no, wait: Wait, \( g(1) = 3^1 -5 = -2 \), \( f(1) = -2 \). Wait, but let's recalculate:

Wait, \( f(1) = |1 - 3| -4 = 2 -4 = -2 \). \( g(1) = 3^1 -5 = -2 \). So at \( x=1 \), \( f(x) = g(x) \)? Wait, but let's check the options. Wait, maybe I made a mistake. Wait, no, let's recheck:

Wait, \( f(1) = |1 - 3| -4 = 2 -4 = -2 \). \( g(1) = 3^1 -5 = 3 -5 = -2 \). So \( f(1) = g(1) \). But the options include "x=1". Wait, but let's check the table again.

Wait, the problem's table:

For \( x=0 \):

\( f(0) = |0 -3| -4 = 3 -4 = -1 \)

\( g(0) = 3^0 -5 = 1 -5 = -4 \)

\( x=1 \):

\( f(1) = |1-3| -4 = 2 -4 = -2 \)

\( g(1) = 3^1 -5 = 3 -5 = -2 \)

\( x=2 \):

\( f(2) = |2-3| -4 = 1 -4 = -3 \)

\( g(2) = 3^2 -5 = 9 -5 = 4 \)

\( x=3 \):

\( f(3) = |3-3| -4 = 0 -4 = -4 \)

\( g(3) = 3^3 -5 = 27 -5 = 22 \)

\( x=4 \):

\( f(4) = |4-3| -4 = 1 -4 = -3 \)

\( g(4) = 3^4 -5 = 81 -5 = 76 \)

Wait, so at \( x=1 \), \( f(x) = g(x) = -2 \). So the solution is \( x=1 \)? But let's check the options. The options are:

  • \( x = 1 \)
  • between \( x=1 \) and \( x=2 \)
  • \( x=2 \)
  • between \( x=2 \) and \( x=3 \)

But according to the table, at \( x=1 \), \( f(x) = g(x) \). So the answer would be \( x=1 \). But let's confirm.

Wait, maybe I made a mistake in \( g(1) \). Let's recheck: \( 3^1 = 3 \), \( 3 -5 = -2 \). \( f(1) = |1-3| -4 = 2 -4 = -2 \). So yes, \( f(1) = g(1) \). So the solution is \( x=1 \).

Part 2: Where does \( f(x) = g(x) \) have a solution?

From the table:

  • At \( x=0 \): \( f(0) = -1 \), \( g(0) = -4 \) → \( f(x) > g(x) \)
  • At \( x=1 \): \( f(1) = -2 \), \( g(1) = -2 \) → \( f(x) = g(x) \)
  • At \( x=2 \): \( f(2) = -3 \), \( g(2) = 4 \) → \( f(x) < g(x) \)
  • At \( x=3 \): \( f(3) = -4 \), \( g(3) = 22 \) → \( f(x) < g(x) \)
  • At \( x=4 \): \( f(4) = -3 \), \( g(4) = 76 \) → \( f(x) < g(x) \)

So the equation \( f(x) = g(x) \) has a solution at \( x = 1 \).

Final Answers:
Table completion:
\( x \)\( f(x) \)\( g(x) \)
1-2-2
2-34…

Answer:

(for the solution part):
A. \( x = 1 \) (assuming the options are labeled, but the problem's options are: "x = 1", "between x = 1 and x = 2", "x = 2", "between x = 2 and x = 3". So the correct answer is \( x = 1 \).