## Subject I. (30 points)

Let the function $f: R \rightarrow(0, \infty), f(x)=\frac{21^{x}+1}{3^{x}+7^{x}}$.

a) Study the injectivity of the function;

b) Is the function f surjective?

c) What is the number of solutions to the equation $f(x)+f(-x)=1$?

Prof. Eugen Jecan, National College "Andrei Mureşanu" Dej
## Subject I.

a) We have $f(-x)=\frac{\frac{1}{21^{x}}+1}{\frac{1}{3^{x}}+\frac{1}{7^{x}}}=\frac{1+21^{x}}{7^{x}+3^{x}}=f(x),(\forall) x \in R$. Therefore, the function is even and thus not injective. (10 points)

b) We will show that $f(x) \geq 1,(\forall) x \in R$. We have $f(x) \geq 1 \Leftrightarrow \frac{21^{x}+1}{3^{x}+7^{x}} \geq 1 \Leftrightarrow\left(3^{x}-1\right)\left(7^{x}-1\right) \geq 0$, which is true for any real number $x$. Therefore, the image of the function is $[1, \infty)$ and thus it is not surjective.

(10 points)

c) Since $f(-x)=f(x)$, we have $2 f(x)=1 \Leftrightarrow f(x)=\frac{1}{2}$, which has no solutions because $f(x) \geq 1,(\forall) x \in R$.

(10 points)