Question

utilice la simetría para resolver problemas
este es el gráfico del primer cuadrante de una función par.
la función es ▼ para -9 < x < -7.
la función es creciente para ▼.
el punto ▼ se encuentra en la gráfica de la función

Explanation:

To solve this problem, we use the property of even functions: an even function satisfies \( f(-x) = f(x) \), which means its graph is symmetric about the \( y \)-axis. Let's analyze each part:

1. Behavior for \( -9 < x < -7 \)

For an even function, the behavior on \( -9 < x < -7 \) is the mirror image of the behavior on \( 7 < x < 9 \) (since \( -x \) maps to \( x \) in the first quadrant). From the graph, on \( 7 < x < 9 \), the function is decreasing (the \( y \)-values decrease as \( x \) increases). By symmetry, on \( -9 < x < -7 \), the function must be increasing (because the left side of the \( y \)-axis mirrors the right side; if the right side is decreasing, the left side (for negative \( x \)) is increasing).

2. Interval where the function is increasing

From the graph in the first quadrant (\( x > 0 \)):

• The function increases on \( 0 < x < 4 \) (since \( y \)-values rise as \( x \) goes from 0 to 4).

By symmetry (for \( x < 0 \)), the function also increases on \( -4 < x < 0 \) (because \( f(-x) = f(x) \), so the left side mirrors the right side).
Thus, the function is increasing on \( -4 < x < 4 \) (combining the left and right intervals of increase).

3. Point on the graph

For an even function, if \( (a, b) \) is on the graph, then \( (-a, b) \) is also on the graph. From the graph, we can see points like \( (4, 9) \) (at \( x = 4 \), \( y = 9 \)). By symmetry, \( (-4, 9) \) must also be on the graph.

Final Answers (assuming dropdown options align with these):

• For \( -9 < x < -7 \): creciente (increasing)
• For where the function is increasing: \( -4 < x < 4 \) (or equivalent interval from symmetry)
• For the point: \( (-4, 9) \) (or similar symmetric point)

(Note: The exact dropdown options are not fully visible, but the reasoning uses the symmetry of even functions to determine behavior and points.)

Answer:

To solve this problem, we use the property of even functions: an even function satisfies \( f(-x) = f(x) \), which means its graph is symmetric about the \( y \)-axis. Let's analyze each part:

1. Behavior for \( -9 < x < -7 \)

For an even function, the behavior on \( -9 < x < -7 \) is the mirror image of the behavior on \( 7 < x < 9 \) (since \( -x \) maps to \( x \) in the first quadrant). From the graph, on \( 7 < x < 9 \), the function is decreasing (the \( y \)-values decrease as \( x \) increases). By symmetry, on \( -9 < x < -7 \), the function must be increasing (because the left side of the \( y \)-axis mirrors the right side; if the right side is decreasing, the left side (for negative \( x \)) is increasing).

2. Interval where the function is increasing

From the graph in the first quadrant (\( x > 0 \)):

• The function increases on \( 0 < x < 4 \) (since \( y \)-values rise as \( x \) goes from 0 to 4).

By symmetry (for \( x < 0 \)), the function also increases on \( -4 < x < 0 \) (because \( f(-x) = f(x) \), so the left side mirrors the right side).
Thus, the function is increasing on \( -4 < x < 4 \) (combining the left and right intervals of increase).

3. Point on the graph

For an even function, if \( (a, b) \) is on the graph, then \( (-a, b) \) is also on the graph. From the graph, we can see points like \( (4, 9) \) (at \( x = 4 \), \( y = 9 \)). By symmetry, \( (-4, 9) \) must also be on the graph.

Final Answers (assuming dropdown options align with these):

• For \( -9 < x < -7 \): creciente (increasing)
• For where the function is increasing: \( -4 < x < 4 \) (or equivalent interval from symmetry)
• For the point: \( (-4, 9) \) (or similar symmetric point)

(Note: The exact dropdown options are not fully visible, but the reasoning uses the symmetry of even functions to determine behavior and points.)