Q131
ArithmeticAlgebra, Equations & Operations
If $f(x) = 5^x + \dfrac{1}{5^x} + 2$, then the value of $f(-x)$ is
- a.$f(x)$
- b.$-f(x)$
- c.$f\left(-\dfrac{1}{x}\right)$
- d.$-f\left(\dfrac{1}{x}\right)$
Answer: (A) $f(x)$
Replace $x$ by $-x$: $f(-x)=5^{-x}+\dfrac{1}{5^{-x}}+2=\dfrac{1}{5^x}+5^x+2$, which is exactly $f(x)$. A function with $f(-x)=f(x)$ is called an even function, and its graph is symmetric about the y-axis; other examples are $x^2$ and $\cos x$. If $f(-x)=-f(x)$ the function is odd, for example $x^3$ and $\sin x$. Here $f(x)$ can never equal $-f(x)$ because it is always positive, and the options involving $\dfrac{1}{x}$ do not simplify to $f(-x)$.