Certainly! Let's go through the problem step-by-step.
### (a) Finding the Zeros of the Function Graphically
The function given is:
[tex]\[ f(x) = \sqrt{2x + 15} \][/tex]
To find the zeros graphically, we would normally use a graphing utility to plot the function and identify the x-values where the function intersects the x-axis. The zero of the function is where [tex]\( f(x) = 0 \)[/tex].
However, since we have already determined the zeros, let's specify them:
[tex]\[ x = -\frac{15}{2} \][/tex]
So the zeros of the function are:
[tex]\[ x = -\frac{15}{2} \][/tex]
### (b) Verifying the Zeros Algebraically
Now, let's verify this result algebraically. We need to solve the equation:
[tex]\[ f(x) = 0 \][/tex]
This translates to:
[tex]\[ \sqrt{2x + 15} = 0 \][/tex]
To solve this, follow these steps:
1. Isolate the square root:
[tex]\[ \sqrt{2x + 15} = 0 \][/tex]
2. Square both sides:
[tex]\[ (\sqrt{2x + 15})^2 = 0^2 \][/tex]
[tex]\[ 2x + 15 = 0 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ 2x + 15 = 0 \][/tex]
[tex]\[ 2x = -15 \][/tex]
[tex]\[ x = -\frac{15}{2} \][/tex]
Thus, we have verified algebraically that the zero of the function is:
[tex]\[ x = -\frac{15}{2} \][/tex]
Combining both parts:
- The zero of the function identified graphically is:
[tex]\[ x = -\frac{15}{2} \][/tex]
- The zero of the function verified algebraically is:
[tex]\[ x = -\frac{15}{2} \][/tex]
Therefore, the answer for both parts (a) and (b) is:
[tex]\[ x = -\frac{15}{2} \][/tex]
