High School

Seja [tex]f[/tex] uma função do [tex]1^{\circ}[/tex] grau tal que [tex]f(2x) = 2f(x)[/tex], definida para qualquer número real. Além disso, [tex]f(9) - f(6) = 18[/tex]. Assim, [tex]f(14)[/tex] vale:

a) 28
b) 42
c) 56
d) 70
e) 84

Answer :

To solve this problem, we start by analyzing the given information about the function [tex]\( f \)[/tex].

1. We know that [tex]\( f \)[/tex] is a linear function, which can be generally written as:
[tex]\[
f(x) = ax + b
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.

2. Given that [tex]\( f(2x) = 2f(x) \)[/tex] for any real number [tex]\( x \)[/tex], we can plug [tex]\( f(x) = ax + b \)[/tex] into this equation:
[tex]\[
f(2x) = a(2x) + b = 2ax + b
\][/tex]
and
[tex]\[
2f(x) = 2(ax + b) = 2ax + 2b
\][/tex]

So,
[tex]\[
a(2x) + b = 2ax + 2b
\][/tex]
Equating the coefficients, we get:
[tex]\[
2ax + b = 2ax + 2b
\][/tex]
By comparing the terms, we find that the constant terms must be equal:
[tex]\[
b = 2b
\][/tex]
Simplifying this, we get:
[tex]\[
b = 0
\][/tex]

Hence, the linear function simplifies to:
[tex]\[
f(x) = ax
\][/tex]

3. We are also given that [tex]\( f(9) - f(6) = 18 \)[/tex]. Substituting the simplified form of [tex]\( f(x) = ax \)[/tex] into this:
[tex]\[
f(9) - f(6) = 9a - 6a = 3a
\][/tex]
Since [tex]\( 3a = 18 \)[/tex], we solve for [tex]\( a \)[/tex]:
[tex]\[
3a = 18
\][/tex]
[tex]\[
a = 6
\][/tex]

Therefore, the function [tex]\( f(x) \)[/tex] can now be written as:
[tex]\[
f(x) = 6x
\][/tex]

4. Finally, we need to find [tex]\( f(14) \)[/tex]:
[tex]\[
f(14) = 6 \times 14 = 84
\][/tex]

Thus, the value of [tex]\( f(14) \)[/tex] is:
[tex]\[
\boxed{84}
\][/tex]

The correct answer is:
e) 84

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