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------------------------------------------------ Given [tex]f(x) = 2x^2 - 4[/tex], which of the following represents a point on the graph of the function?

A. [tex]f(3) = 14[/tex]

B. [tex]f(-5) = 46[/tex]

C. [tex]f(4) = 28[/tex]

D. [tex]f(-2) = -8[/tex]

To determine which of the given points lies on the graph of the function [tex]$ f(x) = 2x^2 - 4 $[/tex], we need to plug each [tex]$ x $[/tex]-value from the options into the function and see if the resulting [tex]$ f(x) $[/tex]-value matches the provided [tex]$ y $[/tex]-value.

Let's evaluate [tex]$ f(x) $[/tex] for each option:

Option A: [tex]$ f(3) $[/tex]
[tex]\[ f(3) = 2(3)^2 - 4 = 2 \cdot 9 - 4 = 18 - 4 = 14 \][/tex]
The result is [tex]$ f(3) = 14 $[/tex], not [tex]$ 16 $[/tex]. This option is incorrect.

Option B: [tex]$ f(-5) $[/tex]
[tex]\[ f(-5) = 2(-5)^2 - 4 = 2 \cdot 25 - 4 = 50 - 4 = 46 \][/tex]
The result matches provided [tex]$ f(-5) = 46 $[/tex]. This option is correct.

Option C: [tex]$ f(4) $[/tex]
[tex]\[ f(4) = 2(4)^2 - 4 = 2 \cdot 16 - 4 = 32 - 4 = 28 \][/tex]
The result matches provided [tex]$ f(4) = 28 $[/tex]. This option is correct.

Option D: [tex]$ f(-2) $[/tex]
[tex]\[ f(-2) = 2(-2)^2 - 4 = 2 \cdot 4 - 4 = 8 - 4 = 4 \][/tex]
The result is [tex]$ f(-2) = 4 $[/tex], not [tex]$ -8 $[/tex]. This option is incorrect.

After evaluating each option, we can conclude that the points that lie on the graph of the function [tex]$ f(x) = 2x^2 - 4 $[/tex] are:

- Option B: [tex]$ f(-5) = 46 $[/tex]
- Option C: [tex]$ f(4) = 28 $[/tex]

Thus, the correct points on the graph of the function are those found in Option B and Option C.