what is the effect of replacing f(x) with f(x - 3)?\na. the range of f(x - 3) is the same as the range of…

Answer:

A. The range of $f(x - 3)$ is the same as the range of $f(x)$.

Explanation:

Step1: Recall function - transformation rule

Replacing $x$ with $x - h$ in $y = f(x)$ gives $y=f(x - h)$, which is a horizontal shift.

Step2: Analyze domain - change

For $y = f(x)$ and $y = f(x - 3)$, the domain of $f(x-3)$ is obtained by solving $x-3\in D_f$ (where $D_f$ is the domain of $f(x)$). So the domain of $f(x - 3)$ is the domain of $f(x)$ shifted 3 units to the right, not the same.

Step3: Analyze range - change

A horizontal shift does not change the set of output values. So the range of $f(x - 3)$ is the same as the range of $f(x)$.

Step4: Analyze x - intercept change

Let the $x$-intercept of $f(x)$ be $x_0$ such that $f(x_0)=0$. For $f(x - 3)$, we set $f(x - 3)=0$, then $x-3=x_0$ or $x=x_0 + 3$. So the $x$-intercept of $f(x - 3)$ is 3 more than the $x$-intercept of $f(x)$.

Step5: Analyze y - intercept change

The $y$-intercept of $f(x)$ is $f(0)$, and the $y$-intercept of $f(x - 3)$ is $f(- 3)$. There is no such relation that the $y$-intercept of $f(x)$ is 3 less than the $y$-intercept of $f(x - 3)$.