Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = 5^x, g(x) = log￬5 x

Verified step by step guidance

1. Step 1: Understand that two functions $f(x)$ and $g(x)$ are inverses if $f(g(x))=x$ and $g(f(x))=x$ for all $x$ in the domains of $g$ and $f$ respectively.
2. Step 2: Start by checking $f(g(x))$. Substitute $g(x)={\log }_{5}x$ into $f(x)=5^x$, resulting in $f(g(x))=5^{{\log }_{5}x}$.
3. Step 3: Simplify $5^{{\log }_{5}x}$ using the property of logarithms that states $a^{{\log }_{a}b}=b$. This simplifies to $x$.
4. Step 4: Now, check $g(f(x))$. Substitute $f(x)=5^x$ into $g(x)={\log }_{5}x$, resulting in $g(f(x))={\log }_5(5^x)$.
5. Step 5: Simplify ${\log }_5(5^x)$ using the property of logarithms that states ${\log }_a(a^b)=b$. This simplifies to $x$.

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If f(x) is a function, its inverse, denoted as f⁻¹(x), satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f⁻¹. This means that applying the function and then its inverse returns the original input.

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where a is a positive constant. They exhibit rapid growth or decay and are characterized by their base raised to a variable exponent. Understanding their properties is crucial for analyzing their inverses, particularly in determining relationships with logarithmic functions.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are expressed as g(x) = log_a(x), where a is the base. They answer the question: to what exponent must the base a be raised to obtain x? The relationship between logarithmic and exponential functions is fundamental in verifying if two functions are inverses of each other.