31. Whichare inverses? (look at the table to answer) $$ f(x)=\frac{1}{10} x \quad g(x)=10^{x} \quad h(x)=10 x \quad j(x)=\log _{10} x $$

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To determine which functions are inverses of each other, let's analyze each pair:

1. **Linear Functions:**
   - $$ f(x) = \frac{1}{10}x $$
   - $$ h(x) = 10x $$

**Check Inverses:**
   - $$ h(f(x)) = 10 \left( \frac{1}{10}x \right) = x $$
   - $$ f(h(x)) = \frac{1}{10} \left( 10x \right) = x $$

Since $$ h(f(x)) = x $$ and $$ f(h(x)) = x $$, **$$ f $$ and $$ h $$ are inverses of each other**.

2. **Exponential and Logarithmic Functions:**
   - $$ g(x) = 10^{x} $$
   - $$ j(x) = \log_{10}x $$

**Check Inverses:**
   - $$ g(j(x)) = 10^{\log_{10}x} = x $$
   - $$ j(g(x)) = \log_{10}(10^{x}) = x $$

Since $$ g(j(x)) = x $$ and $$ j(g(x)) = x $$, **$$ g $$ and $$ j $$ are inverses of each other**.

**Summary:**
- **$$ f(x) $$ and $$ h(x) $$** are inverses.
- **$$ g(x) $$ and $$ j(x) $$** are inverses.

Answer: Both f  and h are inverses of one another, and g is the inverse function of j. Thus f↔h and g↔j.
$$ f(x) $$ and $$ h(x) $$ are inverses of each other, and $$ g(x) $$ and $$ j(x) $$ are inverses of each other.

Whichare inverses? (look at the table to answer)

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