First slide

Theory of expressions

Question

If $x \in R,$ then the expression $9^{x} - 3^{x} + 1$ assumes

Moderate

A

all real values
B

all real values greater than 0
C

all real values greater than $3 / 4$
D

all real values greater than $1 / 4$

Solution

Since the function $f (x) = 9^{x} - 3^{x} + 1$ is continuous for all $x$ and every continuous function attains every value between its maximum and minimum values. Therefore, $f (x)$ takes every value between its minimum and maximum values. We have
$f (x) = 9^{x} - 3^{x} + 1 = {(3^{x} - \frac{1}{2})}^{2} + \frac{3}{4} > \frac{3}{4}$ for all $x$
Thus, $f (x)$ assumes all real values greater than $3 / 4$ .

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