Question

whenever the input of a function $f$ increases by 1, the output increases by 5.
which of these equations could define $f$?
$\bigcirc$ $f(x) = 5^x$
$\bigcirc$ $f(x) = 5x + 3$
$\bigcirc$ $f(x) = 3x + 5$
$\bigcirc$ $f(x) = x^5$

Explanation:

Step1: Identify rate of change condition

The problem states that for every 1-unit increase in input \(x\), the output \(f(x)\) increases by 5. This means the function has a constant rate of change (slope) of 5, so it is a linear function of the form \(f(x)=mx+b\) where \(m=5\).

Step2: Analyze each option

• For \(f(x)=5^x\): This is an exponential function, its rate of change is not constant.
• For \(f(x)=5x+3\): This is linear with slope \(m=5\). If \(x\) becomes \(x+1\), \(f(x+1)=5(x+1)+3=5x+5+3=f(x)+5\), which matches the condition.
• For \(f(x)=3x+5\): This is linear with slope \(m=3\), so output increases by 3 when input increases by 1, which does not match.
• For \(f(x)=x^5\): This is a polynomial function, its rate of change is not constant.

Answer:

\(f(x) = 5x + 3\)