If the function $f$ satisfies $4^{x} + 4^{f (x)} = 4$ , then the domain of the function $f$ is

(1, \infty)

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(0, 1)

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(- 1, \infty)

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(- \infty, 1)

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Solution

The correct option is D $(- \infty, 1)$
$4^{x} + 4^{f (x)} = 4$
$\Rightarrow 4^{f (x)} = 4 - 4^{x}$
Taking $log$ with base $4$ both sides
$\Rightarrow f (x) = {log}_{4} (4 - 4^{x})$
For $log$ to be defined
$\Rightarrow 4 - 4^{x} > 0 \Rightarrow 4 > 4^{x} \Rightarrow 1 > x ∴ x \in (- \infty, 1)$

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If the functionf satisfies 4x+4f(x)=4, then the domain of the function f is

The correct option is D (−∞,1)4x+4f(x)=4 ⇒4f(x)=4−4x Taking log with base 4 both sides ⇒f(x)=log4(4−4x) For log to be defined ⇒4−4x>0⇒4>4x⇒1>x∴x∈(−∞,1)

If the function $f$ satisfies $4^{x} + 4^{f (x)} = 4$ , then the domain of the function $f$ is

The correct option is D $(- \infty, 1)$
$4^{x} + 4^{f (x)} = 4$
$\Rightarrow 4^{f (x)} = 4 - 4^{x}$
Taking $log$ with base $4$ both sides
$\Rightarrow f (x) = {log}_{4} (4 - 4^{x})$
For $log$ to be defined
$\Rightarrow 4 - 4^{x} > 0 \Rightarrow 4 > 4^{x} \Rightarrow 1 > x ∴ x \in (- \infty, 1)$